Learn Dutch: Ik laat mijn handen masseren. - I am having my hands massaged.

In English, we say we "have" something done by someone else. In Dutch, you use the verb laten (which literally means "to let" or "to leave") to express that you are arranging for someone else to do an action for you.

Why is it masseren (to massage) instead of a past word like "gemasseerd" (massaged)?

English uses a past participle here ("massaged"), but the Dutch laten structure requires a bare infinitive (masseren). Even though it looks like an active verb, Dutch speakers understand from this structure that your hands are receiving the massage, not giving it!

Why does masseren go at the very end of the sentence?

This is standard Dutch word order for sentences with two verbs. The conjugated verb (laat) takes the second position in the sentence, and the infinitive verb (masseren) is pushed all the way to the end.

If the subject was "he" (hij) instead of "I" (ik), how would laat change?

It wouldn't change at all! The stem of the verb is laat. Because Dutch spelling rules forbid ending a word with a double "t", the form for "he", "she", or "you" is simply laat. So "He is having his hands massaged" is Hij laat zijn handen masseren.