Jeg drikker et glas vin til maden.

Because jeg is the subject form, meaning I. It is the person doing the action.

• Jeg drikker ... = I drink / I am drinking ...
• Mig means me and is used as an object, not as the subject of a normal statement.

Drikker is the present tense of drikke (to drink).

A very useful thing about Danish is that the verb usually has the same form for all persons in the present tense:

• jeg drikker
• du drikker
• han/hun drikker
• vi drikker

So the -er here is not specifically because of jeg; it is just the normal present-tense form.

It can mean both.

Danish often uses the simple present where English might use either:

• I drink
• I am drinking

The exact meaning depends on context. So Jeg drikker et glas vin til maden can mean something like:

• I drink a glass of wine with my meal (habitual), or
• I am drinking a glass of wine with the meal (right now)

Because glas is a neuter noun in Danish, so it takes et in the indefinite singular.

Danish nouns are mainly divided into two grammatical genders:

• en-words
• et-words

So:

• et glas = a glass

You simply have to learn the gender of each noun as part of the word.

Because vin is being used as a mass noun inside the expression et glas vin.

In Danish, just like in English, you normally say:

• a glass of wine
• et glas vin

You do not need an extra article before vin here. The word vin tells you what the glass contains.

Because maden is the definite form of mad.

• mad = food / meal
• maden = the food / the meal

Danish usually puts the definite article at the end of the noun as a suffix, instead of using a separate word like the.

So:

• mad = food
• maden = the food / the meal

It can mean either, depending on context. In this sentence, maden is most naturally understood as the meal.

So til maden usually means something like:

• with the meal
• with dinner
• at mealtime

In everyday Danish, mad often refers not just to food in general, but to the meal you are having.

Here til is best understood as with or along with in the sense of served with a meal.

Literally, til often means to, but prepositions do not always match English directly. In this sentence, til maden is an idiomatic Danish way to talk about a drink that goes with the food.

So:

• vin til maden = wine with the meal

Sometimes, but til maden is the more natural and idiomatic choice here.

• til maden suggests something served with the meal, especially drinks
• med maden can sound more literally together with the food

So a Danish speaker would very naturally say:

• Jeg drikker et glas vin til maden

The sentence follows the normal Danish main-clause pattern:

• Jeg = subject
• drikker = finite verb
• et glas vin = object
• til maden = prepositional phrase

So the structure is:

Subject + verb + object + extra information

That is the basic word order in a simple statement like this.

Yes. That is also correct, but the meaning is slightly less specific.

• Jeg drikker vin til maden = I drink wine with the meal
• Jeg drikker et glas vin til maden = I drink a glass of wine with the meal

The version with et glas emphasizes one glass. The version without it is more general.

Neither in the usual countable sense. Here vin is a mass noun, like water in English.

So Danish treats it as something you do not normally count directly. You usually count the container or amount instead:

• et glas vin = a glass of wine
• en flaske vin = a bottle of wine

That is why vin stays in its basic form here.