Just like in real football, each year fantasy football leagues have a draft (note: in dynasty leagues, this normally consists of NFL rookies only), in which each team drafts NFL players. These players are kept unless "dropped" (aka become free agents) or are traded. In most leagues, no player may be owned by more than one team, (although some leagues do allow for this).
There are essentially two types of drafts. In a traditional "serpentine" (aka "snake") draft, owners take turns drafting players in a "serpentine" method, i.e. the owner who picks 1st in the odd rounds picks last in the even rounds, in the interests of fairness. In an auction draft, each owner has an (imaginary) budget which he must use to purchase all his players in an auction format. Owners take turns nominating players for open bid. The owner who bids the highest on each player receives that player (reducing their remaining budget accordingly). A few leagues use a hybrid of the two styles, selecting a portion of their roster via auction, with the remainder selected through a serpentine method