Generally speaking the four terms could be defined as follows.

Mandates - These were awarded to victorious powers after WW1 by the League of Nations (forerunner of the United Nations), for example France in Syria, Britain in what was then called the Mandate of Palestine. Ostensibly these were territories that had broken free of Ottoman rule but were thought incapable of governing/protecting themselves so Allied powers were called in as guardians. I'm not really qualified to speak at length at how benign or exploitative this 'guardian' relationship may have been, however most political maps would identify mandates the same way as any other sovereign territory held by an imperial power. These mandates had little, if any independence. The term 'Mandate' is still in use today to describe United Nations peacekeeping missions, although obviously these have a different character than the old, Post-WW1 mandates.

Protectorates - Protectorates are similar to Mandates in that they're ostensibly weak or less advanced nations that require the assistance of a great power for economic, or military purposes. Generally they'd have some degree of local autonomy but their foreign affairs were handled by the protecting power. The distinction here is that protectorates weren't awarded by the League of Nations, they were formed (more or less) bi-laterally.

Dominions - This is an exclusively British term to refer to their former colonies which had achieved a degree of independence but still acknowledged British supremacy. In WW2 these were (essentially) Canada, Australia, New Zealand, and South Africa although Ireland, Newfoundland, and sometimes the then-British government of India have also been characterized as Dominions. Keep in mind that being a Dominion isn't just a matter of historical identification, it brings with it a degree of dependency on Britian and in the case of Canada, Australia, New Zealand (unsure on South Africa) it also means your Head of State is the British monarch. Therefore the United States of America was never considered a 'Dominion'. In modern times, there is a trace of this relationship in the 'Commonwealth of Nations' which is essentially an exclusive club for former British colonies and territories that form an extremely loose political and economic bloc.

Colonies - A Catch-all term for any non-European territory that an Imperial power held. It's also not mutually exclusive with the other terms - the British Dominions could all be considered 'Colonies'. I've also heard the French and British Mandates called 'Colonies' as well. This term doesn't necessarily mean that large amounts of immigration from the motherland to the colony in question took place, although of course there was usually at least a small amount.

Follow up. What's a territory as well in this context? How was say, Puerto Rico viewed as a territory and not a protectorate?