It means to be born of the spirit as well as of the flesh

When we believe that Jesus Christ is the way the truth and the life, then we also believe that God is his true father.
By believing this we receive of his spirit, which is created within us and we are now born again of incorruptible [spiritual] seed.

1Pe 1:23    Being born again, not of corruptible seed, but of incorruptible, by the word of God, which liveth and abideth for ever.

This spiritual seed given us cannot be removed nor destroyed because it is incorruptible. By the word of God it is given and so therefore it will be everlasting. Just as the father and his son are also everlasting. In our flesh we may fall time and time again from Gods word, and therefore need Gods help in restoring us back to our walk with him, which began when we first knew him.  Jesus himself spoke of this being born again and what it meant:

John 3:1-8
1    There was a man of the Pharisees, named Nicodemus, a ruler of the Jews:
2    The same came to Jesus by night, and said unto him, Rabbi, we know that thou art a teacher come from God: for no man can do these miracles that thou doest, except God be with him.
3    Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again [first in the womb, then again - (which means in the Greek - {from above}], he cannot see the kingdom of God.
4    Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother's womb, and be born?
5    Jesus answered, Verily, verily, I say unto thee, Except a man be born of water [in the womb] and of the Spirit [God's spiritual nature], he cannot enter into the kingdom of God.
6    That which is born of the flesh is flesh [in the womb]; and that which is born of the Spirit is spirit.
7    Marvel not that I said unto thee, Ye must be born again [born of spirit as well as of flesh in the womb].
8    The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit.

Nicodemus was presenting a very non spiritual understanding of the thing Jesus was telling him. Jesus however was speaking of a future prophesy which would come to pass on the day of Pentecost - when people could now be born of God.  When we are born in water in the womb, we are considered born of the flesh - in order for us to be born again, we must first have been born in the womb. When we are born of God it is not of the flesh, but of the spirit. The spirit which cannot be seen nor felt by us in our present bodies but it is created within us at the moment of the new birth by God. This new creation gives us the ability to spiritually communicate with our heavenly father, by revelation and more. At Christ's appearing this spirit within us will blossom into it's fullness, and we will then be given a new spiritual body. Then we will know things in their full true light and not clouded by our limited earthly vision.  See "Some difficult to understand scriptures" link for a better understanding of verses 5 and 6.

1Co 13:12    For now we see through a glass, darkly; but then face to face: now
I know in part; but then shall I know even as also I am known.

Now our vision is clouded by many things, because we are earthly by our nature. Yet then [at his return] we will see the heavenly things. Now we know in part: The scriptures, the world, and our flesh, among many other things. Then things will be different as shown below.

1Jo 3:2    Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is.

When we see him, how will he appear to us? This I cannot fully answer but only guess. There are very few references in the word to describe how Christ appeared in his resurrected body to others. To them he appeared as a man. However at his appearing we will be physically changed to be like he is in all aspects.

Luke 24:15-53
15    And it came to pass, that, while they communed together and reasoned, Jesus himself drew near, and went with them.
16    But their eyes were holden that they should not know him.
17    And he said unto them, What manner of communications are these that ye have one to another, as ye walk, and are sad?
18    And the one of them, whose name was Cleopas, answering said unto him, Art thou only a stranger in Jerusalem, and hast not known the things which are come to pass there in these days?
19    And he said unto them, What things? And they said unto him, Concerning Jesus of Nazareth, which was a prophet mighty in deed and word before God and all the people:
20    And how the chief priests and our rulers delivered him to be condemned to death, and have crucified him.
21    But we trusted that it had been he which should have redeemed Israel: and beside all this, to day is the third day since these things were done.
22    Yea, and certain women also of our company made us astonished, which were early at the sepulchre;
23    And when they found not his body, they came, saying, that they had also seen a vision of angels, which said that he was alive.
24    And certain of them which were with us went to the sepulchre, and found it even so as the women had said: but him they saw not.
25    Then he said unto them, O fools, and slow of heart to believe all that the prophets have spoken:
26    Ought not Christ to have suffered these things, and to enter into his glory?
27    And beginning at Moses and all the prophets, he expounded unto them in all the scriptures the things concerning himself.
28    And they drew nigh unto the village, whither they went: and he made as though he would have gone further.
29    But they constrained him, saying, Abide with us: for it is toward evening, and the day is far spent. And he went in to tarry with them.
30    And it came to pass, as he sat at meat with them, he took bread, and blessed it, and brake, and gave to them.
31    And their eyes were opened, and they knew him; and he vanished out of their sight.
32    And they said one to another, Did not our heart burn within us, while he talked with us by the way, and while he opened to us the scriptures?

Jesus revealed to these men the scriptures; concerning his crucifixion, death, and resurrection. Then for a moment they finally realized who he was, but then he was gone from their sight.

33    And they rose up the same hour, and returned to Jerusalem, and found the eleven gathered together, and them that were with them,
34    Saying, The Lord is risen indeed, and hath appeared to Simon.
35    And they told what things were done in the way, and how he was known of them in breaking of bread.
36    And as they thus spake, Jesus himself stood in the midst of them, and saith unto them, Peace be unto you.
37    But they were terrified and affrighted, and supposed that they had seen a spirit.

What spirit did thy see? They saw Jesus in his resurrected body, that's who.

38    And he said unto them, Why are ye troubled? and why do thoughts arise in your hearts?
39    Behold my hands and my feet, that it is I myself: handle me, and see; for a spirit hath not flesh and bones, as ye see me have.
40    And when he had thus spoken, he shewed them his hands and his feet.
41    And while they yet believed not for joy, and wondered, he said unto them, Have ye here any meat?
42    And they gave him a piece of a broiled fish, and of an honeycomb.
43    And he took it, and did eat before them.
44    And he said unto them, These are the words which I spake unto you, while I was yet with you, that all things must be fulfilled, which were written in the law of Moses, and in the prophets, and in the psalms, concerning me.
45    Then opened he their understanding, that they might understand the scriptures,
46    And said unto them, Thus it is written, and thus it behoved Christ to suffer, and to rise from the dead the third day:
47    And that repentance and remission of sins should be preached in his name among all nations, beginning at Jerusalem.
48    And ye are witnesses of these things.
49    And, behold, I send the promise of my Father upon you: but tarry ye in the city of Jerusalem, until ye be endued with power from on high.
50    And he led them out as far as to Bethany, and he lifted up his hands, and blessed them.
51    And it came to pass, while he blessed them, he was parted from them, and carried up into heaven.
52    And they worshipped him, and returned to Jerusalem with great joy:
53    And were continually in the temple, praising and blessing God. Amen.

John 20:1-31
1    The first day of the week cometh Mary Magdalene early, when it was yet dark, unto the sepulchre, and seeth the stone taken away from the sepulchre.
2    Then she runneth, and cometh to Simon Peter, and to the other disciple, whom Jesus loved, and saith unto them, They have taken away the Lord out of the sepulchre, and we know not where they have laid him.
3    Peter therefore went forth, and that other disciple, and came to the sepulchre.
4    So they ran both together: and the other disciple did outrun Peter, and came first to the sepulchre.
5    And he stooping down, and looking in, saw the linen clothes lying; yet went he not in.
6    Then cometh Simon Peter following him, and went into the sepulchre, and seeth the linen clothes lie,
7    And the napkin, that was about his head, not lying with the linen clothes, but wrapped together in a place by itself.
8    Then went in also that other disciple, which came first to the sepulchre, and he saw, and believed.
9    For as yet they knew not the scripture, that he must rise again from the dead.
10    Then the disciples went away again unto their own home.
11    But Mary stood without at the sepulchre weeping: and as she wept, she stooped down, and looked into the sepulchre,
12    And seeth two angels in white sitting, the one at the head, and the other at the feet, where the body of Jesus had lain.
13    And they say unto her, Woman, why weepest thou? She saith unto them, Because they have taken away my Lord, and I know not where they have laid him.
14    And when she had thus said, she turned herself back, and saw Jesus standing, and knew not that it was Jesus.
15    Jesus saith unto her, Woman, why weepest thou? whom seekest thou? She, supposing him to be the gardener, saith unto him, Sir, if thou have borne him hence, tell me where thou hast laid him, and I will take him away.
16    Jesus saith unto her, Mary. She turned herself, and saith unto him, Rabboni; which is to say, Master.
17    Jesus saith unto her, Touch me not; for I am not yet ascended to my Father: but go to my brethren, and say unto them, I ascend unto my Father, and your Father; and to my God, and your God.
18    Mary Magdalene came and told the disciples that she had seen the Lord, and that he had spoken these things unto her.
19    Then the same day at evening, being the first day of the week, when the doors were shut where the disciples were assembled for fear of the Jews, came Jesus and stood in the midst, and saith unto them, Peace be unto you.
20    And when he had so said, he shewed unto them his hands and his side. Then were the disciples glad, when they saw the Lord.
21    Then said Jesus to them again, Peace be unto you: as my Father hath sent me, even so send I you.
22    And when he had said this, he breathed on them, and saith unto them, Receive ye the Holy Ghost:
23    Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.
24    But Thomas, one of the twelve, called Didymus, was not with them when Jesus came.
25    The other disciples therefore said unto him, We have seen the Lord. But he said unto them, Except I shall see in his hands the print of the nails, and put my finger into the print of the nails, and thrust my hand into his side, I will not believe.
26    And after eight days again his disciples were within, and Thomas with them: then came Jesus, the doors being shut, and stood in the midst, and said, Peace be unto you.
27    Then saith he to Thomas, Reach hither thy finger, and behold my hands; and reach hither thy hand, and thrust it into my side: and be not faithless, but believing.
28    And Thomas answered and said unto him, My Lord and my God [this is a figure of speech - representing humbleness before one greater than oneself - Jesus was not the creator of heaven and earth to Thomas, but rather his master [Jesus] whom he had great reverence toward (people still use similar figures of speech today without even a second thought)].
29    Jesus saith unto him, Thomas, because thou hast seen me, thou hast believed: blessed are they that have not seen, and yet have believed.
30    And many other signs truly did Jesus in the presence of his disciples, which are not written in this book:
31    But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name.

When Jesus appeared to the disciples he did not look like a ghost or any other spiritual creature.  He simply looked like the man who they knew him to be, yet while also showing them [the disciples - especially Thomas (who doubted that this man was truly Jesus)] the scars from his crucifixion to help them to believe who he was.