Groups: On the course level you can assign a user to one (or more) groups. In a course you can assign a context(activity) to a group. When members of the group leave the course, they lose their identity with the group.
Groupings are a collection of groups. If you had 10 groups, you could combine or mix and match the smaller groups into 2 or more groupings. You can assign a grouping to a context like an assignment.
Cohorts:On the site or category level you assign a user to a cohort. Cohorts are used to bulk enrol users into one or more courses. If you wish the cohort to be members of a group within the course you will then have to add them to a group in a further step.