As the other four major leagues in American sports (NBA, NHL, MLB, MLS) the NFL follows the draft system in which college players sign up to be chosen by teams. NFL Draft Picks is a term used to refer to those choices (picks) that were made, or are going to be made by teams. It is common to hear phrases such as "The Seahawks have 5 picks in this year's NFL draft". That means that the Seahawks will have five opportunities to choose players during the draft. The term "Draft pick" can also be used to refer to a player chosen by a team during the draft. It is common to hear stuff like "Kyler Murray was the Cardinals' first draft pick in 2019."
The NFL draft has 7 rounds consisting of 32 picks per round, one for each team. The choosing order is decided by teams' overall record on the previous season, meaning that the team that finishes last (the one with the worst record) gets the first pick on every round and the Super Bowl champion gets the last pick on every round. But that doesn't mean that the choosing order will follow league's standings as teams can trade their draft picks away for players or even future draft picks.
The round and number of pick in which a player was selected will follow him through his career but that doesn't mean that players picked in the first round will be good or that players picked in the seventh round won't be good. Tom Brady, one of the best Quarterbacks in history was chosen in the 6th round of the 2000 NFL draft. A player's journey in the NFL only starts at the draft, and the hardest part is still to come.
After being picked, rookies (as first-year players are called) sign their "rookie deals," which are designated by the league. There is no fixed amount that Draft Picks get paid; the league uses a formula to calculate it. Rookies get to attend the "rookie minicamps," where they get to feel for the first time how NFL teams function.