Learn Hardcore Swedish: Han väntar på sin bror vid stationen. - He is waiting for his brother at the station.

In Swedish, when you say that you are waiting for someone or something, you almost always use vänta på plus the thing or person: vänta på tåget, vänta på min vän.
Vänta on its own is more like just wait in a place or wait in general: Vi väntar här (We will wait here).
So to match the English meaning wait for his brother, Swedish needs the preposition på.

What exactly does på mean here? Is it like English for?

Literally, på often corresponds to English on, at, or in, but here it is part of the fixed verb phrase vänta på.
You should think of vänta på as a single unit meaning wait for.
Even though the English preposition is for, you cannot translate prepositions one-by-one; Swedish simply uses på with vänta when there is an object.

Why is it sin bror and not hans bror?

Sin is the reflexive possessive pronoun; it refers back to the subject of the clause.
So Han väntar på sin bror means that he is waiting for his own brother.
Hans bror would usually mean someone else’s brother (his brother, but not the subject’s own), which would be a different meaning.

How do I know when to use sin, sitt, or sina?

They all mean his/her/its/their own and refer back to the subject, but they agree with the thing possessed:

sin is used with an en-word in singular: sin bror, sin bok
sitt is used with an ett-word in singular: sitt hus, sitt barn
sina is used with any noun in plural: sina bröder, sina böcker, sina hus

In this sentence, bror is an en-word in singular, so you must use sin.

Why is there no article before sin bror, like en or den?

In Swedish, you normally do not use an article together with a possessive.
You say min bror, din vän, hans bil, sin bror, not en min bror or den hans bil.
The possessive word (min, din, sin, hans, hennes, etc.) itself makes the noun specific, so no separate article is needed.

Why is väntar in the simple present, when English would usually say is waiting?

Swedish does not have a separate progressive tense like English is waiting.
The simple present väntar can mean both waits and is waiting, depending on context.
So Han väntar på sin bror naturally corresponds to English He is waiting for his brother here.

Why is it vid stationen and not på stationen or i stationen?

Vid literally means by or next to and emphasizes that he is near the station, at or by its side.
På stationen is also very common and usually means at the station, often with a neutral or slightly more general sense of being at that location (inside or on the premises).
I stationen focuses more on being physically inside the building itself.
So vid stationen highlights the idea of being by the station; på stationen would also be a natural choice in many contexts.

Why is it stationen (with -en) and not just station?

The -en ending makes station definite, so stationen means the station.
Swedish often uses the definite form where English uses the with a common noun:

huset = the house
boken = the book
stationen = the station

Here, a specific station is meant, so the definite form stationen is used.

Can vid stationen appear in other places in the sentence?

Yes. Basic Swedish word order is quite flexible for adverbials like vid stationen, as long as the verb stays in second position in main clauses.
Some natural alternatives are:

Han väntar vid stationen på sin bror.
Vid stationen väntar han på sin bror.

The meaning remains essentially the same; moving vid stationen changes the emphasis more than the content.

What is the difference between bror and broder, and what is the plural?

Bror is the normal everyday word for brother.
Broder is a longer form that sounds more formal, old-fashioned, or poetic; you might see it in fixed expressions or older texts.
The usual plural is bröder (brothers), and the definite plural is bröderna (the brothers).
So: en bror – två bröder – bröderna.