Learn Italian: La signora aspetta suo marito. - The lady is waiting for her husband.

The verb aspettare has the "for" built right into its meaning. English says "to wait for someone," but Italian just says "to wait someone." Adding a word like per (for) after the verb would be incorrect here.

Does aspetta mean "waits" or "is waiting"?

It means both! In Italian, the regular present tense covers both general habits ("she waits") and actions happening right now ("she is waiting").

Since it is "her" husband, why is it the masculine suo instead of the feminine sua?

In Italian, the possessive word always matches the gender of the person or thing being owned, not the owner. Since marito (husband) is masculine, you must use the masculine suo, even though the owner (la signora) is a woman.

Why is it just suo marito and not il suo marito?

You normally need a "the" (definite article) before a possessive, like in la sua macchina (her car). However, you drop the article when talking about singular family members. Since marito is a singular family member, it is simply suo marito.