Learn Turkish: Akşam yemeğini annem yapıyor.

Akşam yemeği means evening meal or dinner. The base word is yemek (food or meal), but it changes to yemeği to link it with akşam (evening). Then, because dinner is the specific thing being made (the direct object), we must add the accusative case ending -i. The -n- in the middle is simply a buffer letter used to separate the two vowels.

Why is annem (my mother) in the middle of the sentence instead of at the beginning?

The standard word order in Turkish is Subject-Object-Verb, which would be Annem akşam yemeğini yapıyor. However, Turkish allows you to move words around to change the focus. The word placed immediately before the verb gets the most emphasis. By putting annem right before the verb, the speaker is emphasizing that their mother is the one making dinner, as opposed to someone else.

Why use yapıyor for cooking? Doesn't it usually mean doing or making?

Yes, the root verb yapmak means to do or to make. Just like in English where we say making dinner, it is very common and natural in Turkish to use yapmak when talking about preparing a meal. You could also use the specific verb for cooking (pişiriyor), but yapıyor is perfectly correct here.

Where is the word for my in this sentence?