Learn Russian: Не обижай младшего брата.

You are learning reflexive verbs like обижаться (to take offense / feel hurt). However, this sentence uses the non-reflexive form обижать (to offend / hurt someone else). If you said не обижайся, it would mean "don't be offended" rather than "don't hurt someone."

Why does младшего брата look like it's in the Genitive case?

It is actually in the Accusative case because the brother is the direct object of the verb. As a quick refresher: for masculine animate nouns (people and animals), the Accusative case endings perfectly copy the Genitive endings (-а for the noun and -его for the adjective).

Why use the imperfective verb обижай? Isn't it a specific command?

In Russian, negative commands (telling someone not to do something) almost always take the imperfective aspect, even if it is a single, specific situation. That is why we use the imperfective не обижай instead of a perfective equivalent.

Does младшего mean "little" or "younger" here?

Literally, младший means "younger." But in Russian, it is the standard way to refer to a "little" brother or sister, so both translations are correct depending on how you would naturally say it in English.