How did Elizabeth lose her small finger?

Hover over to see the solution (spoiler-alert):

Booker sold her to Comstock when she was a baby. Just before the portal closed, baby-Elizabeth stuck her pinky out of the portal reaching for her father (Booker) and it got cut off as it closed.

Her finger (or lack of it) is the most important part of Bioshock Infinite. In fact, it's what made the game possible.

Spoiler alert:

You see, Comstock is the baptized Dewitt. After baptism, he eventually develops Columbia. He hires the female physicist Lutece, which develops a trans-dimensional device. Once she developed it, she brings herself as "himself" from another dimension. Comstock realizes he's sterile. So, he needs someone to continue his quest of cleansing the sodom below (burn down the USA to "purify it"). He knows he will die before that happens, so he tells Lutece to go back in time and get her baby Anna (Elizabeth) back in the day when he was in his office as a detective. Dewitt was so much in debt from gambling that he sold his daughter to Lutece to "wipe away the debt". The male Lutece picks up baby Elizabeth (Anna). He is eventually chased down by Dewitt. Dewitt finds Comstock and male Lutece in a back alley with the baby ready to cross a portal. They both (male Lutece and Comstock) cross with the baby. In the struggle, there is a fraction of delay from when the female Lutece shuts down the machine and the portal closes. As we know, the butterfly effect of the struggle between Comstock and Dewitt causes part of the finger of the baby to take longer to cross the portal. The portal closes and slices the finger off. Because of that, the universe and all universes (millions of millions per Elizabeth) are out of balance and the equation doesn't add up. The result is that the baby Elizabeth is now a "glitch" and that is what enables her to open or generate tears. This was an unplanned event by Comstock, that is why they had to develop a funnel, and this had to be done before her raging teen hormones started at menarche and she seeks independence (she could go anywhere). Without the ability to generate tears, she is confined to the tower. And all because of the finger.

The rest, well, you already know. The full explanation is much longer but I didn't want to bore you.

SPOILERS BELOW

In her Voxophone log, "The Source of Her Power", Rosalind Lutece says:

What makes the girl different? I suspect is has less to do with what she is, and rather more with what she is not. A small part of her remains from where she came. It would seem the universe does not like its peas mixed with its porridge.

At the end of the game, you will learn that Booker DeWitt sold his baby (Anna) to Comstock (who was assisted by Robert and Rosalind Lutece), to pay for his gambling debts. He tried to stop Comstock from taking his baby in the last minute, but failed to do so. As the tear closed, Anna's little finger was caught in the closing tear and was cut. Rosalind Lutece says that this is how Elizabeth (which is what Comstock named Anna as) got her powers.