Check into means to register to stay:
I'm going to check into the hotel.
I'm going to check into the hospital.
Check in can be used when there's no object. This still refers to staying at a hotel.
I need to go check in first, then we can go out to eat.
Check in and check into have various other meanings as well.
check in (with someone) = to let someone know where you are or where you are going to be (or what you will be doing)
check in (something) = to look or verify something somewhere
check into (something) = to investigate something