How can the roommate that signed the lease get rid of the other roommate that is just an occupant of the apartment?
If this roommate pays you rent or was supposed to pay you, then you in effect are their landlord (and they are your "subtenant"). Since you have the legal right to occupy the premises, you can file an "unlawful detainer" action (i.e. eviction proceeding). However you will have to comply with all legal requirements in order to get them lawfully removed. As a general rule this starts with written notice of 30 or so days. If your roommate fails to leave by the date specified, then you will need to go to court. If an unlawful detainer is granted, this person will either have to leave the premises voluntarily or have the sheriff remove them. In the meantime do not undertake any "self-help" measures such as changing the locks, etc. This will work against you. What you should do now is to contact a real estate attorney. They can advise you of the correct way in which to go about this in MO.
Note: If this roommate has a tenancy with the landlord, then the landlord will have to evict them. Such a tenancy could have been established by the roommate paying rent directly to the landlord, the landlord putting (or allowing the roommate to put) their name on the mailbox, etc. A signed lease is not necesary.