There are six kinds of operating system, but since you just asked for two, I'll be happy to be able to mention only two.
1. Single and multi-tasking - single tasking can run one program at a specific time while multi-tasking can run multiple programs in one session.
2. Single and multi-user - single user have no facilities to recognized users but cna run multiple programs at a time while multi-user goes beyond multi-tasking with facilities.