A prison is like a jail, but it's where criminals get put for longer periods of time, usually after they have been convicted in a trial. So when you first get arrested, you're put in a jail. Later, you're moved to a prison.
There are different levels of prison, for people who have done different kinds of crimes. The worst criminals go to a "maximum security prison". This type of prison has high walls, lots of guards, and strict rules. This is to keep any of the prisoners from escaping. "Minimum security" prisons, on the other hand, are less strict. Prisoners who committed less serious crimes, and prisoners who have behaved well for several years at a higher-security prison, go there.