Whoever signs the lease as a tenant is responsible for all of the rent (unless the lease provides otherwise). This means that if three people decide to rent an apartment together, but only one of them signs the lease, then only that one person is responsible to the landlord for all of the rent. Likewise, if all three sign the lease, then each one is responsible for all of the rent. The landlord, of course, cannot collect the same rent more than once, but the landlord is not limited to looking to each roommate for only one-third of the rent.
A roommate who is stuck paying the entire rent may sue any non-paying roommates for their share. It is important to understand that the disagreement is between the roommates, not with the landlord. If a roommate moves out, the tenant should ask the landlord to remove the roommate's name from the lease and get any roommate's name added. Remember, when one roommate moves and others stay, the absent roommate is not relieved from potential liability for all of the rent, if that person signed the lease, unless the landlord agrees.