Each State will have a website, e.g. Texas Boys State.
It's run by the American Legion throughout the nation. I'm a parent, I went to Boys State a LONG time ago. I lived near a military base. We had lots of American Legion posts nearby. They sponsored 5-10 boys every year to go to Boys State. That's a lot more slots than most high schools.
These days, it can be a pretty competitive process to get a slot, depending on your school and how well known the program is. Some schools don't participate at all in sending kids.
Anyhow, if your school has never sent someone, I suggest going to the website and making an inquiry about attending. They might direct you to a nearby American Legion for an application. They might let you pay yourself if you are a good candidate and they want to create future interest at your school. In Texas last year, the cost to attend was about $275.00 (excluding your travel to/from).
If your school has sent people in the past, contact your counselor and see what the application process is.
It's a fantastic program. It's generally about a week long. And, in that short time, you form mock local, county, and state governments.
Being selected and attending adds points to the leadership portion of your Whole Candidate Score. I believe this is because the selection process is generally competitive, the candidates chosen are generally good students involved in numerous activities, and the program is a known quantity to service academies because it is basically uniformly administered across the 50 states by the American Legion.