Learn Hardcore Russian: Мама заставляет брата мыть посуду после ужина. - Mom makes my brother wash the dishes after dinner.

Because брата is in the accusative case, used for the direct object of the verb.
The subject is мама (nominative), and the person who is being made to do something is the direct object: (Кого?) брата.
So the structure is: Мама (who?) заставляет брата (whom?) мыть посуду.
With most masculine nouns ending in a consonant, the accusative form is the same as the genitive: брат → брата.

Why is the verb мыть in the infinitive and not моет?

The verb заставлять normally takes the pattern:

заставлять + (accusative person) + infinitive

This is very close to English “make someone do something”:

Мама заставляет брата мыть посуду. = Mom makes brother wash the dishes.

You do not conjugate the second verb; you keep it in the infinitive:

not заставляет брата моет, but заставляет брата мыть.

Why is it после ужина and not после ужин?

The preposition после always takes the genitive case.
The genitive of ужин is ужина, so you must say после ужина (after dinner).

Pattern: после + genitive

после работы (after work)
после урока (after the lesson)
после ужина (after dinner)

Why is мыть (imperfective) used here, and not помыть?

Мыть is imperfective; помыть is perfective.

Imperfective is used for:

repeated or habitual actions,
processes without focus on completion.

Here the sentence describes what usually happens: mom habitually makes him wash the dishes after dinner. That’s a repeated, routine action, so мыть (imperfective) is natural.

If you wanted to emphasize one single completed act, you could say, for example:

Мама заставила брата помыть посуду вчера. – Mom made her brother wash (and finish washing) the dishes yesterday.

Why is there no word for “her” in брата? Shouldn’t it be её брата?

Russian often omits possessive pronouns (мой, твой, её, его, etc.) with close family members when the owner is obvious from context.

In this sentence, it’s clear that мама and брат belong to the same family, so её is not needed.
Compare:

Мама заставляет брата мыть посуду. – Mom makes (her) brother wash the dishes.
Моя мама заставляет моего брата мыть посуду. – My mom makes my brother wash the dishes. (more explicit)

What exactly does заставляет mean? Is it more like “makes” or “forces”?

Заставлять usually implies some pressure or lack of willingness, stronger than a simple request.
It can correspond to both make and force, depending on tone and context:

neutral / everyday control: Mom makes him wash the dishes.
stronger, emphasizing unwillingness: Mom forces him to wash the dishes.

If you want a softer idea (just asking), you could use просить (to ask):

Мама просит брата мыть посуду. – Mom asks (her) brother to wash the dishes.

What tense and aspect is заставляет, and how would I say “will make”?

Заставляет is:

present tense,
3rd person singular,
imperfective aspect (from заставлять).

The perfective partner is заставить.
To say “will make” in a one-time, future sense, you’d usually use the perfective future:

Мама заставит брата мыть посуду. – Mom will make (her) brother wash the dishes.

If you want a general, repeated future action, you can keep the imperfective:

Мама всегда будет заставлять брата мыть посуду. – Mom will always make her brother wash the dishes.

Can the word order be changed, for example: Мама заставляет мыть посуду брата после ужина?

Russian word order is flexible, but not all orders are natural or unambiguous.
Мама заставляет брата мыть посуду после ужина is the normal, clear order:

мама – subject
заставляет – verb
брата – the person being made to do something
мыть посуду – what he must do
после ужина – when

A sentence like Мама заставляет мыть посуду брата после ужина sounds awkward and can be misread as if брата is connected with посуду.
Best to keep брата directly after заставляет in this kind of structure.

Could this sentence mean “Mom, make brother wash the dishes after dinner”?

No. In modern standard Russian, мама here is nominative, so it is the subject, not a person being addressed.

“Mom, make brother wash the dishes after dinner” would be:

Мама, заставь брата мыть посуду после ужина.

Here:

Мама, with a comma, is a form of address.
заставь is the imperative (command) form of заставить.

How would I negate this sentence: “Mom doesn’t make brother wash the dishes after dinner”?

You simply add не before заставляет:

Мама не заставляет брата мыть посуду после ужина.
= Mom does not make (her) brother wash the dishes after dinner.

The rest of the structure stays the same:
не + заставляет + (accusative person) + infinitive.