Min søster er dygtig til at lave mad.

Because søster is a common-gender singular noun.

Danish possessives change according to the gender and number of the noun being possessed:

• min = for singular common gender
• mit = for singular neuter
• mine = for plural

So:

• min søster = my sister
• mit hus = my house
• mine søstre = my sisters

In Danish, when you use a possessive like min, din, hans, vores, etc., you normally do not use an article as well.

So Danish says:

• min søster = my sister

not

• min en søster

This is similar to English, where we say my sister, not my a sister.

Er is the present tense of at være = to be.

So min søster er ... means my sister is ...

In this sentence, er links the subject min søster with the description dygtig til at lave mad.

Dygtig means something like:

• skilled
• capable
• good at
• competent

In this sentence, dygtig describes your sister’s ability.

A useful nuance:

• dygtig often sounds a bit more like skilled/competent
• god often sounds more general, like good

So dygtig til at lave mad is very natural and means she is skilled at cooking.

After dygtig, Danish often uses til to mean at in the sense of good at.

So:

• dygtig til noget = good at something
• dygtig til at gøre noget = good at doing something

Examples:

• Hun er dygtig til matematik. = She is good at math.
• Han er dygtig til at synge. = He is good at singing.

So in your sentence:

• dygtig til at lave mad = good at cooking

Because lave is in the infinitive form here.

• at lave = to make / to cook

After dygtig til, Danish can use an infinitive verb phrase:

• dygtig til at lave mad
• god til at skrive
• svær at forstå

The at here is the normal infinitive marker, like English to in to cook.

Yes, literally it is make food, but idiomatically it means cook.

So:

• at lave mad = to cook

This is a very common everyday expression in Danish.

You can think of it as:

• literal structure: make food
• natural English meaning: cook

Because mad here is being used in a general sense, like food or cooking.

So:

• at lave mad = to cook / to make food

You are not talking about one specific dish or one specific meal. It is the general activity of cooking.

If you were talking about something specific, the wording could be different depending on the context.

Yes, absolutely. That is also natural Danish.

The difference is mostly nuance:

• god til = good at
• dygtig til = skilled / competent at

In many everyday situations, they are very close in meaning.
Dygtig can sound a little more focused on ability or skill.

It follows a very common Danish pattern:

• Min søster = subject
• er = verb
• dygtig = subject complement / adjective
• til at lave mad = phrase explaining what she is good at

So the structure is basically:

subject + verb + adjective + complement

This is very similar to English:

My sister + is + good + at cooking

This is a word many learners find tricky.

A rough guide:

• dyg- has a vowel sound somewhat like the vowel in French u or German ü, which English does not really have
• the g is not pronounced like a strong English g
• -tig is often pronounced very lightly

A practical learner-friendly approximation might be something like:

• DÜG-ti
• or DUHG-ti with a very soft middle sound

But the exact sound is hard to capture in English spelling. The most difficult part is the Danish y in dygtig.

You would change er to var:

• Min søster var dygtig til at lave mad.

That means:

• My sister was good at cooking.

The rest of the sentence stays the same.