Learn Italian: Questa è un'università moderna.

Because università is a feminine noun that starts with a vowel. The feminine word for "a" is una, but when the next word starts with a vowel, it drops its final 'a' and uses an apostrophe to flow better, becoming un'. If it were a masculine noun starting with a vowel, you would just use un without an apostrophe.

Why do we say Questa instead of Questo here?

In Italian, the word for "this" must match the gender of the noun it refers to. Since università is a feminine noun, we use the feminine singular form Questa.

Why does moderna come after the noun instead of before it like in English?

Unlike English, Italian descriptive adjectives usually come right after the noun they describe. Also, notice that moderna ends in 'a' so that it correctly matches the feminine singular noun università.

What does the accent mark on the last letter of università do?

It tells you to put the speaking stress on that very last syllable (oo-nee-ver-see-TAH). As a helpful bonus rule, Italian nouns that end in an accented vowel like this one do not change their spelling when they become plural.