The draft order is based on how the 32 teams finished the previous season. The worst team (Rams) picks first based on the worst record. The Super Bowl Champ (Saints) will pick 32nd. After each team selects a player that completes a round. The draft has a total of 7 rounds. Once selected, players agents negotiate a contract for the rookies. Although players can't refuse to be drafted by a team, there have been a couple of situations in which the player made it known that he would not play for a certain team if they drafted him (John Elway, Eli Manning).
That is just a basic idea of how it works. There are other circumstances that can come in to play such as teams trading draft picks and the NFL awarding teams extra draft picks in the later rounds based on the amount of free agents they lost the year prior.
Hope that explains it for you.