European basketball players can apply for draft, just as any other player in world (USA colleague or high school player). Most of them are already scouted by some teams.
All NBA teams participate in draft lottery. There is decided in what order what team will pick players. NBA team with worst record last season has biggest chance to win rights to choose players first.
When teams know what pick they have, then they focus on that talent group. If team has mid-range pick (15th, for example), they will try to estimate what group of players will be left by then. So, week before draft night training camps takes place for all players that are to be drafted, and each team focus on particular group of players. They evaluate their skills, plus and cons, scout an evaluate them.
At the draft night, each team chooses player that they think is best from the pool of players left in the order decided by draft lottery.
By this time European players had same treatment as any other player. He was in training camp, teams got time to meet, see and evaluate him. If they choose to pick him, they only get exclusive rights to offer him a contract. But, if he already has good contract in Europe, he will probably not play in NBA for 3 more years.
The reason for this is that player has to buyout his existing contract. Since he is restricted to rookie minimum for first 3 years, NBA team give him enough money to buy his Euro contact. After 3 years, NBA allows bigger contracts, and that is probably the right time for Euro stars to come in NBA.
Good example of this is Nikola Mirotić from Real Madrid. Bulls picked him in 2011 NBA draft (21st overall), but they will not even try to offer him any contract until 2014, when he is no longer restricted to rookie minimum.