Category: an Activity used to categorize the Statement. "Tags" would be a synonym. Category SHOULD be used to indicate a profile of xAPI behaviors, as well as other categorizations. For example: Anna attempts a biology exam, and the Statement is tracked using the cmi5 profile. The Statement's Activity refers to the exam, and the category is the cmi5 profile.
Grouping: an Activity with an indirect relation to the Activity which is the Object of the Statement. For example: a course that is part of a qualification. The course has several classes. The course relates to a class as the parent, the qualification relates to the class as the grouping.
Other: a contextActivity that doesn't fit one of the other properties. For example: Anna studies a textbook for a biology exam. The Statement's Activity refers to the textbook, and the exam is a contextActivity of type other.
Parent: an Activity with a direct relation to the Activity which is the Object of the Statement. In almost all cases there is only one sensible parent or none, not multiple. For example: a Statement about a quiz question would have the quiz as its parent Activity.