The White Rabbit seems to give this "role" to a girl that he chooses, and challenges the "Alice" candidates to participate in a game called "Kill The White Rabbit", in which the White Rabbit is the target himself. The place where he would meet the candidates to explain the rules of the game was in Wonderland's Forest. So far, the candidates have come to the number of eighty-nine, implying that the game is not an easy game to complete. The 89th Alice is the first one to be male, and also the protagonist of the series.
"Alice"-candidates were all somehow infatuated by the name "Alice" for various unknown reasons, and were implied to have "lost their dreams" prior to arriving in Wonderland. In the case of the 88th Alice, it is told how she hated her real name and was happy to have the name "Alice" as her name.
The failed candidates take the form of "Regrets", clinging to the name "Alice". The reason for them turning into Regrets are unknown, but it is strongly implied that they were killed.
Early in the series, the White Rabbit was seen taking care of a girl looking like Alice, who never spoke back to him or even moved from where she sat upon a throne. He called her "Marianne". Based on her looks and her importance to the White Rabbit, she appears to have a connection to the "Alice".