Eviction is the removal of a tenant from rental property by a law enforcement officer; before a tenant can be evicted, a landlord must win an eviction lawsuit. Of course, in meerkat behavior, it's the process of rejecting and removing, a subordinate female from the group by the dominant female. All meerkat groups are started by evicted females, usually a gang of sisters.
Reasons for eviction
Most of the time, females are evicted from the group for giving birth to or just conceiving pups, a right usually reserved for the dominant female. It is to protect the dominant female's pups, which the victim of eviction may kill if she is not removed. 
Another reason for eviction is rebelling or assuming leadership, which most meerkats don't do, but are sometimes forced to; for example, a female in a splinter group without the queen must assume leadership.
On far rarer occasions, males, usually the dominant, will evict other males. This is likely to enforce his own leadership and rid the group of male competition.