Ich winke meiner Mutter am Bahnhof.

Because winken normally takes the dative case for the person you wave to.

• meine Mutter = nominative or accusative
• meiner Mutter = dative

So in this sentence, meiner Mutter means to my mother, even though English does not show that case ending.

In German, winken often means to wave to someone, and the person is put in the dative:

• jemandem winken = to wave to someone

So:

• Ich winke meiner Mutter = I wave to my mother

German does not need a separate word like to here.

Yes. Ich winke meiner Mutter zu is also correct and very common.

There are two closely related patterns:

• jemandem winken
• jemandem zuwinken

So these are both possible:

• Ich winke meiner Mutter am Bahnhof.
• Ich winke meiner Mutter am Bahnhof zu.

Both mean essentially the same thing. Learners often expect a word corresponding to English to, and in German that can appear as zu with zuwinken, but it is not required with plain winken.

am is a contraction of an dem.

• an dem Bahnhof → am Bahnhof

Here it means something like:

• at the train station
• by the train station

It gives the location where the action happens.

Because this sentence describes a location, not movement toward a destination.

German prepositions like an can take different cases:

• dative for location: wo? = where?
• accusative for direction: wohin? = to where?

Here the sentence means the waving happens at the station, so German uses dative:

• am Bahnhof = an dem Bahnhof

If you were talking about movement toward the station, then accusative could appear:

• Ich gehe an den Bahnhof. = I go to the station area / toward the station

winke is the 1st person singular present tense of winken.

The verb is conjugated like this in the present tense:

• ich winke
• du winkst
• er/sie/es winkt
• wir winken
• ihr winkt
• sie/Sie winken

So Ich winke means I wave or sometimes I am waving, depending on context.

No, the word order can change, but the conjugated verb still has to stay in second position in a normal statement.

This sentence has a neutral order:

• Ich winke meiner Mutter am Bahnhof.

You could also say:

• Am Bahnhof winke ich meiner Mutter.

That puts more emphasis on the location. Because meiner Mutter is marked by the dative case, German can move things around more freely than English.

Because meiner is already doing the job of a determiner.

In meiner Mutter:

• meiner = my in the feminine singular dative form
• Mutter = mother

German normally does not use both a possessive word and an article together in this kind of phrase. So you say:

• meiner Mutter

not:

• der meiner Mutter or anything similar

Because all nouns are capitalized in German.

So in this sentence:

• Ich is capitalized because it starts the sentence
• Mutter is capitalized because it is a noun
• Bahnhof is capitalized because it is also a noun

This is a basic but very important spelling rule in German.