What is a push in betting?
A push is a tie with the book. The result lands exactly on the number, nobody wins the bet, and you get your stake back.
Landing on the number
Say you take a favorite at -7 and it wins 24 to 17. It won by 7, no more and no less, so neither side of the bet has won. You laid $110 to win $100, and the book hands back the $110. A total works the same way: at 44, a 24 to 20 final is a push for the over and the under alike.
The book keeps nothing on a push, not even the vig. Apart from a void, it is the only way a bet can end with the house taking nothing from anybody.
Why some lines can push and some can't
Only a whole number can push. Put the spread at -7.5 and the favorite either wins by 8 or it doesn't, so somebody always wins. That half point is called the hook, and books hang it on a lot of lines for exactly that reason.
In football the push is a real outcome, not a technicality, because games end on margins of 3 and 7 more often than on any others (the spreads page goes into why). If I'm laying -3, I count the push in before I compare it with -2.5 somewhere else at a higher price. The half point is worth the most right there.
A tie on a two-way moneyline is usually a push as well, at most books. A three-way line is different: the draw is its own result, so a bet on either team loses when the game ends level. If you want your money back on a draw, that is the bet called draw no bet, which the three-way moneylines page explains. (At a blackjack table the word means the same thing: you tie the dealer and keep your bet.)
On a parlay or a teaser
Most books drop a pushed leg off a parlay and pay the rest as if the leg had never been there, so a three-team parlay with a push is paid as a two-teamer. Printed cards can be different, and some say ties lose. Teasers vary more again: at some books a pushed leg drops off, at some a two-teamer with a push is refunded, and at some a tie loses.
I'd read those rules before the bet, not after the push. The house rules page covers what most books do, and your own book's page is the one that counts.
A push won't make you money. It won't cost you any either, which is more than most bets can say.
Related questions
What does the spread mean in betting?
The spread is a head start the book gives the underdog to make the game an even bet. A team at -7 has to win by more than 7 for a bet on it to win. A team at +7 wins the bet if it wins the game or loses by fewer than 7.
What does over/under mean?
The over/under is a bet on the combined score of both teams. The book posts a number, called the total, and you bet whether the final score adds up to more or less than it. Who wins the game doesn't matter.
What is a parlay?
A parlay is one bet made of two or more picks. Every pick has to win for the ticket to pay, and the payout multiplies with each one you add. Miss one and the whole ticket loses.