A push in sports betting occurs when the point spread or point total hits the exact number in which you bet. For example, you can bet the Dallas Cowboys as a -7 point favorite to beat the Green Bay Packers or the Packers as a +7 point underdog. If the Cowboys win by seven points, both bets are a push and money is refunded (you get your money back). Similarly, if you also bet the total (over/under) in that game which was 47 points and it finished 27-20, that bet would also be pushed.
While it's more common in football and basketball, pushes are possible in almost every sport. The over/under for a baseball game could be set at seven runs and it's a similar case for soccer and hockey.
If you're betting on a parlay or teaser, it's a little more complicated and often depends on the rules of the sportsbook. If you have a four-team parlay and one of your bets was the Cowboys -7 which pushed, your parlay would drop down to three teams. However, each sportsbook has their own rules and there are places that will count a push in a parlay or teaser, as a loss. While that isn't common, it's sometimes worth looking into the rules prior to betting at a certain sportsbook.