Alice has the power to see into the future. She can focus in on certain people or search for certain events. Her visions aren't perfect, as the
final decisions regarding the future she sees have to have been made at the time of her viewing them. She can't see a future very well that hasn't been decided yet. This makes predictable things like the weather easy for her to read. She also can't see anything involved with the werewolves/shape-shifters or half vampires.
In Twilight, Alice used her ability to predict when the storm would be happening so her family could play baseball.
Alice also could see James in Bella's old dance studio in Phoenix, and Bella later being captured by him,once Bella made the decision to meet him there.
In New Moon, Alice was able to see Edward's decision to provoke the Volturi but could not see exactly how he would do it because he kept changing his mind.
In Breaking Dawn, Alice's ability allowed her to warn her family about the confrontation with the Volturi.