Users with check user rights have access to a special page which can assist with investigation into some forms of abuse, in particular detection of "sock puppets" (i.e. a single user with multiple accounts). Sock puppets are not specifically banned and can be useful, but should not be used to create a false consensus (or, even worse, a false argument).
The check user tool can show the IP addresses used by a particular user account, the user accounts using a particular IP address or address range and the edits made from a particular IP address or address range. Use of this tool has privacy implications so all uses are logged. However, this log is only visible to users with check user rights as the log itself will contain information which may have privacy implications.