Amar is a minor character from Allegiant whom we get to know a little better in these prequel stories. Amar is Four's initiation trainer. He is to Four what Four is to Tris later on, but without the kissing. (Although we doubt Amar would object if the opportunity came up. More on that later.)
Amar is the driving force for many of Four's biggest moments. Amar trains Four, but also tries to convince him to develop a more rounded social life. He's also the one who suggests that Four is Divergent (even though he's wrong about that!)
When Four starts showing "abnormal" (2.283) test results, Amar hides them. Unfortunately, he's not as good at hiding his own abnormal results, and he gets "killed" as a result. We put killed in quotes because we learn in Allegiant that Amar is still alive. His death, like that of Four's mother, Evelyn, was faked.
Aside from seeing the training process a bit, we get little hints at Amar's sexuality. Early on he "eyes [Tobias]" seeming "more curious than anything else" (1.206). (Can you blame him?) And later on we see a boy who "sits slouched against Amar like they're old friends, or brothers" (2.21). Or more?