As noted in other answers, the Greek ἄνωθεν [anōthen] means either again or above. As C.K. Barrett in his commentary states:
ανωθεν is capable of two meanings and here it probably has both. It may mean "from above", but also "afresh", "again". The birth which is required here is certainly a second birth, but it is not (see v. 4) a mere repetition of man's first birth, but a begetting from above, from God.1
Since Jesus uses the word twice, each meaning can be applied:
A person must be born again to see the Kingdom of God
A person must be born from above to enter the Kingdom of God
This sequence elaborates on the central theme of the Prologue:
But to all who did receive him, who believed in his name, he gave the right to become children of God, who were born, not of blood nor of the will of the flesh nor of the will of man, but of God. (John 1:12-13 ) [ESV throughout]
A person has the authority to become a child of God. This authority is not from earthly rights or privileges. It is from the will of God. People who become children of God are born again from (God) who is above.
The focal point of Jesus' teaching Nicodemus is rebirth as children of God. Later Jesus teaches there is a different type of rebirth:
He said to them, “You are from below; I am from above (ἄνω). You are of this world; I am not of this world. (John 8:23)
First, Jesus is from above ἄνω [anō] which means from on high or of time, formerly. As with anōthen both meanings apply. Jesus is anō, from above and from before.2
Next, the others are from below. The word here is κάτω which means below or beneath either of place or temporal succession [katō]. As with anōthen and anō both meanings can be applied. The others have been "reborn" from below:
You are of your father the devil, and your will is to do your father's desires... (John 8:44)
This explains the distinction Jesus made with Nicodemus. There is a difference between seeing and entering the Kingdom of God. Seeing the kingdom of God does not mean entrance, because those who are from below also see the Kingdom of God, but their perspective is negative. It something they oppose:
And if I cast out demons by Beelzebul, by whom do your sons cast them out? Therefore they will be your judges. But if it is by the Spirit of God that I cast out demons, then the kingdom of God has come upon you. (Matthew 12:27-28)
Therefore the complete teaching to Nicodemus is:
Jesus answered him, “Truly, truly, I say to you, unless one is born again [either from above or from below] he cannot see the kingdom of God.” (3:3)
Every person born again [either from above or below] will see the Kingdom of God
Jesus answered, “Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God. That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. Do not marvel that I said to you, ‘You (plural) must be born from above.’ The wind blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit.” (3:5-8)
Only people born of water and Spirit from above will enter the Kingdom of God
Notes: 1. C.K. Barrett, *The Gospel According to St. John*, S.P.C.K., 1962, pp. 171-172 2. The word *anōthen* used in the discourse with Nicodemus, is from ἄνω used in John 8. Thus Jesus is **not** from above *anōthen* (again/above). Rather Jesus is from above *anō* (before/above).