Learn Russian: Мой брат хочет уехать навсегда.

Both use the у- prefix meaning departure, but уехать specifically means to leave by transport. This is the natural verb to use when talking about moving away or going on a long trip. уйти means to leave on foot, like walking out of a room or building.

Since this is something he "wants" to do, why do we use the perfective verb уехать instead of the imperfective уезжать?

We use the perfective уехать because the brother wants to complete the action once and achieve the final result (leaving and being gone). If you used the imperfective уезжать, it would sound like he wants to leave on a regular basis, or simply wants to be in the process of leaving.

How is the word навсегда built?

The word навсегда is built by combining the preposition на (for) and the word всегда (always). It literally translates to "for always," which is how Russian says "forever."

Why is уехать in its dictionary form here instead of changing to match Мой брат?

When you use two verbs together for a single idea, only the first one changes to match the subject. Here, the first verb хочет (wants) is already conjugated to match брат. The second verb уехать stays in the infinitive (dictionary) form, exactly like the "to leave" part in English.