The Premier League (and football leagues in general) use a point system. 3 points for a win, 1 point for a draw and 0 points for a loss. The team that is top of the table come the end of the season is crowned the champion. The top four teams qualify for the Champions League and the bottom three teams are relegated to the division below (in this case the Championship). There's also the Capital one Cup and the FA Cup. Promotion in the lower leagues (Championship, League One, League , etc) is a bit different. In the Championship, the top two teams are promoted to the Premier League while the teams from 3rd-6th compete in the playoffs to determine who gets the third promotion slot.
There's also the Europa League, but I won't get into that.