Every round in a modern NFL Draft features 32 picks—one pick provided for each of the league's 32 franchises in each round. Every team should have seven choices coming into a draft before accounting for trades, picks awarded, compensation provided, or penalties levied.
Remember that teams can gain draft picks in rare instances where, say, a team loses a minority executive or assistant coach to become a general manager or head coach for another team. Teams can also lose picks if, for instance, they are penalized for cheating or participating in some fraudulent activity (e.g. the Chiefs tampering early with free agent Jeremy Maclin cost them a third-round choice).
Teams are allowed to trade picks up to three years in advance and four "draft cycles" in advance once the current year's draft has already started. Compensatory picks are also allowed to be traded once they've been made official shortly before the draft.
The number of compensatory picks awarded in a given year isn't a fixed number. The amount will vary based on a formula that takes into account the free-agent losses and gains of each franchise from the previous year. The exact details of the formula are not publicly disclosed (cue NFL conspiracy theories!), but it considers factors such as the contracts of departing free agents, the playing time and performance of those players with their new teams, postseason honors for that player, and the overall net loss or gain of qualifying free agents by each team.
A maximum of 32 compensatory picks is awarded each year, spread across the end of the third through seventh rounds of the first-year player draft held at the end of April. However, not all teams receive comp picks, and the number awarded to each team can vary.