Elizabeth's power was that she could see all the different possibilities. What she was doing at the end was to make sure that all possibilities had the same result. That Booker ALWAYS died. No matter what happened.
The game deals with mainly with alternate timelines. Even when you "die" in the game, when you see the door of your office, what that is implying is that Booker actually died. His story ended at that death. What happens next is that you're playing now as an ALTERNATE booker in a different timeline where everything that happened up until the point of the game loading up was exactly the same as in the previous timeline, except this Booker survived up until that point. It's also why enemies are given back health and you lose money. Because they're technically not the same enemies and this alternate Booker has collected a slightly less amount of money than the now-dead one in the other timeline.
What Elizabeth does at the end is to find the one point where all the different timelines branch off, and to just kill him. Make sure there are no other possible ways for it to go.
The ending, however, leaves a big question mark on that.
But I have a particular theory about the game: Elizabeth knows that she's only stopping one possible Booker and stopping one possible branch of choices, but there still are the possible C and D and E universes.
The C and D and E are further playthroughs. Every time you play the game, you're actually a different timeline's Booker. And each Elizabeth in those different timelines comes to the same conclusion and same solution to the problem. However, the title of the game reflects the amount of times Elizabeth would have to kill Booker to make sure, once and for all, there will be no Comstock: infinite.
And the game can be played an infinite number of times, each time slightly different, because it's a new timeline each time. A new Elizabeth, a new Booker.