Off the tourist track: A visit to the Criminal Courthouse

One way to see the Criminal Court House is to jump the turnstile. The other way is to visit. No appointment, no need to get permission as it is open to the public. But you will have to go through the security booth screening like everyone who enters the courthouse.
The New York Criminal Courthouse is located at 100 Centre Street in downtown Manhattan. and is open 7 days a week, Monday - Wednesday: 8:00 a.m. to 1:00 a.m. and Thursday - Sunday: 9:00 a.m. to 1:00 a.m. (yes, 1 in the morning!). Find info at Criminal Court.
After you get through the security booth, ask any of the court officers what courtrooms have "Arraignments".  This is where people who have been arrested in the last 24 hours see the judge and the prosecutor for the first time. The process is very quick, you probably wont know what is going on, but it is much more easy to get an idea of the "system" than sitting in one trial that might not have anything going on at the time.
* Turn off your cell phone BEFORE you go in. When you are in the courtroom, do NOT sit in the first 2 rows. Do NOT talk. The judge will get annoyed, and may ask you to leave. Or worse...throw you in the slammer!