Here is how it goes:
Thousands audition for low-level producers. The good, bad and unique are selected to go on to the next round.
The second round is in front of the executive producers. This is where they are questioned about their “backstory” in more detail. If they are good, bad or interesting enough they then go on to the judges.
The judges then choose 100-200 to go to Hollywood.
In Hollywood, there are a series of cuts. In most recent years, they sing individually, with groups, and again individually. They are then divided into rooms. The rooms have random numbers of people, from about 15-35 in the past. About 40-45 people are selected to go on.
After the Hollywood round, the judges and producers review the tapes and choose 12 men and 12 women to go on to the semifinals. They try to balance for different types of singers, ethnicities, and ages.
The top 12 men and top 12 women sing on subsequent nights and the audience votes. For three weeks, 2 men and 2 women are voted off each week until 6 men and 6 women remain for the finals.
In the finals, all contestants perform and one is voted off each week.
In prior years, there were 30-32 semifinalists who competed in heats with the top 2 or 3 in each group moving on to the finals. This led to an uneven balance of men and women and they had to institute a wildcard and judges’ picks to balance it. Also, after voting for a contestant in the first heat, the audience would not see him/her until 4 weeks later. Thus the change to the current system where men and women compete separately until 6 of each is chosen.
FWIW, getting into the top 12 is less important than getting into the top 10, because the top 10 only go on tour.