There are three common meanings for "push" in poker:
Push is common poker slang for going all-in. It is named because a player will sometimes literally push all their chips into the center of the table in one motion.
Sometimes, "push" is used in combination with the "all-in" term (as in "She pushed all-in once she hit the turn"), but it can also be used with no qualifiers to indicate going all-in ("On the turn, she pushed and everyone else folded").
Push can also refer to a poker dealer moving to a new table at the beginning/end of a down (you may hear dealers say "I'm waiting for my push and then I'm out of here!" or "My push is late - is anyone going to push me?").
Push is also the term used in gambling to indicate a tie, most common in table games such as blackjack. If a player holds the same hand as the dealer, a push is declared and the player has her bet returned; she neither wins nor loses. Occasionally, players will use the term in poker to describe a split pot where they chop it with another player, since that is also a tie - but this usage is rare.