Learn Italian: Ho ventun anni. - I am twenty-one years old.

In Italian, you don't say that you are a certain age; you say that you have a certain number of years. Because ho means I have, the sentence literally translates to: I have twenty-one years.

Shouldn't twenty-one be ventuno? Why is it ventun here?

Yes, the number on its own is ventuno. But when a number ending in -uno comes right before a masculine noun like anni, it drops the final -o to become ventun. This follows the exact same pattern as the word for a/an, which is un before most masculine words.

Since the number ends in one, shouldn't the word for years be singular?

No, because the total amount is still more than one. Just like in English, twenty-one is a plural amount, so it must be followed by the plural noun anni (years), not the singular anno (year).

How do you combine venti (twenty) and uno (one) to make ventuno?

When combining a tens number like venti (twenty) or trenta (thirty) with uno (one) or otto (eight), you drop the last vowel of the tens word. So venti and uno join together to become ventuno (not ventiuno).