Learn Russian: Мама варит борщ.

The verb варить specifically means "to boil" or "to cook in liquid". In Russian, you usually use варить when making soups, broths, or boiling vegetables, whereas готовить is the general word for "to cook" or "to prepare".

Since борщ is the direct object (the thing being made), shouldn't its ending change?

It is actually in the accusative case here! However, because борщ is a masculine, inanimate noun (a thing, not a living person or animal), its accusative form looks exactly the same as its nominative (dictionary) form.

How do we know this means "Mum is making" and not "Mum makes"?

Russian only has one present tense. Варит can mean either "is making" (happening right now) or "makes" (happening regularly). Without extra time words like сейчас (now) or каждый день (every day), you just figure it out from the context of the conversation.

Can I change the word order to Борщ варит мама?

Yes, Russian word order is very flexible! However, the word at the very end of a Russian sentence usually carries the most new information or emphasis. Борщ варит мама would mean "It is Mum who is making the borscht" (answering the question "Who is making the borscht?"). The original sentence is a simple, neutral statement.