Most of the answers are based on the US system, so let me give a very brief explination of how ATC training generally works in Europe.
Overall, you will not get accepted into the training programme unless there is a job for you in the other end. This means that the selection rounds prior to starting training is not just a matter of being fit for training - it also includes a job interview.
Many European countries make use of the First European Air Traffic Controller Selection Test (FEAST), which is a standardized, computer-based test used to evaluate potential ATCO trainees.
FEAST is used to assess your cognitive and mental skills such as IQ, working under time pressure, multi-tasking, logical planning, mathematics etc. etc.
If you pass the FEAST tests (there are two of them), you will be invited to the job interview, which is a pretty standard interview. Following the interview, if you are among the few lucky ones, you will have to undergo a medical examination and security clearance.
If you make it this far, you can start the actual education.
Training to become an ATCO takes about 3 years. The first half is part class-room, part simulator training, focusing on the huge amount of theoretical material you have to know as an air traffic controller. You will find many topics resemble those of pilot training: air law, radio communication, human factors and performance etc., but there are also more technical classes that deal directly with air traffic management.
The second half of the education consists of on-the-job-training (OJT). During this period, you will handle real air traffic in a real environment, under the supervision of an instructor, who will brief, de-brief and generally assist you. You will learn to apply your theoretical knowledge in practise, and get familiar with the airspace and local procedures of the facility where you will end up working.
Of course, during both the first and second half of education, you will have to take tests and exams to ensure you are keeping up with the goals and requirements.
Then finally, after about three years, you will walk into your familiar training facility, but the instructor will be gone, and you just get on with it . . .