Mam dwadzieścia lat.

That is simply the normal Polish pattern for talking about age.

• English uses to be: I am twenty years old
• Polish uses mieć (to have): Mam dwadzieścia lat

So in Polish, age is expressed as something you have, not something you are.

More examples:

• Mam osiemnaście lat = I am 18
• Ona ma trzydzieści lat = She is 30
• Ile masz lat? = How old are you? literally How many years do you have?

Mam means I have.

It is the 1st person singular form of the verb mieć (to have).

A few useful forms:

• mam = I have
• masz = you have
• ma = he/she/it has
• mamy = we have
• macie = you have (plural)
• mają = they have

So:

• Mam dwadzieścia lat = I am twenty years old
• Masz dwadzieścia lat = You are twenty years old

Because Polish usually does not need subject pronouns when the verb already shows who the subject is.

In mam, the ending already tells you it means I have, so ja (I) is usually omitted.

• Mam dwadzieścia lat = normal, natural
• Ja mam dwadzieścia lat = also correct, but ja adds emphasis or contrast

For example:

• Ja mam dwadzieścia lat, a on ma dziewiętnaście.
  I am 20, and he is 19.

So the version without ja is the most neutral one.

Dwadzieścia means twenty.

It is the number telling us how many years old the speaker is.

In this sentence:

• mam = I have
• dwadzieścia = twenty
• lat = years

So the structure is literally: I have twenty years

This is one of the most important things to learn with ages in Polish.

Polish uses different forms of year depending on the number:

• 1 → rok
• 2, 3, 4 → lata
• 5 and more → lat

Examples:

• Mam jeden rok. = I am one year old.
  (In real life, for a person, people more often say Mam rok.)
• Mam dwa lata. = I am two years old.
• Mam trzy lata. = I am three years old.
• Mam cztery lata. = I am four years old.
• Mam pięć lat. = I am five years old.
• Mam dwadzieścia lat. = I am twenty years old.

Also note:

• numbers ending in 2, 3, 4 often use lata, but
• the teens (12, 13, 14) use lat, not lata

So:

• 22 lata
• but 12 lat

Yes. Lat is used as the form of year/years after many numbers in age expressions.

For learners, the most useful rule is practical rather than deeply grammatical:

Use:

• rok after 1
• lata after 2, 3, 4 (except the teens)
• lat after 5+, 0, and the teens

Examples:

• 21 lat
• 24 lata
• 25 lat
• 11 lat
• 14 lat
• 32 lata
• 37 lat

You do not need to master every grammar detail immediately; the main thing is to learn the pattern.

Yes, Polish word order is more flexible than English, but Mam dwadzieścia lat is the most normal and neutral order.

Possible variations:

• Mam dwadzieścia lat. = neutral
• Dwadzieścia lat mam. = marked, poetic, or contrastive
• Ja mam dwadzieścia lat. = emphasizes I

As a beginner, the safest choice is: Mam dwadzieścia lat.

A simple English-friendly approximation is:

mahm dvah-DJESH-cha laht

A few pronunciation notes:

• mam = like mahm
• dw in dwadzieścia is pronounced like dv
• dzie sounds roughly like dje
• ścia sounds roughly like sh-cha
• lat has a clear a like in father, not the English cat

A rough breakdown:

• mam
• dwa-dzieś-cia
• lat

If you want a more careful approximation: mam dva-JE-sh-cha lat

Use the same structure:

Mam + number + rok/lata/lat

Examples:

• Mam rok. = I am one year old.
• Mam dwa lata. = I am two years old.
• Mam dziewięć lat. = I am nine years old.
• Mam piętnaście lat. = I am fifteen years old.
• Mam dwadzieścia jeden lat. = I am twenty-one years old.
• Mam trzydzieści dwa lata. = I am thirty-two years old.
• Mam czterdzieści lat. = I am forty years old.

This is one of the most useful sentence patterns in Polish.

The standard question is:

Ile masz lat? = How old are you?

Literally, it means: How many years do you have?

Other versions:

• Ile ma lat? = How old is he/she?
• Ile macie lat? = How old are you? (to more than one person)
• Ile mają lat? = How old are they?

A short dialogue:

• Ile masz lat?
• Mam dwadzieścia lat.

Yes. It is a perfectly complete and natural sentence.

It already includes:

• the verb: mam
• the number: dwadzieścia
• the noun: lat

You do not need anything else.

You could add extra information if you want:

• Mam dwadzieścia lat i studiuję. = I am twenty years old and I study.
• Mam już dwadzieścia lat. = I am already twenty years old.
• Mam tylko dwadzieścia lat. = I am only twenty years old.

But on its own, Mam dwadzieścia lat. is fully correct.