Learn Italian: Compri alcune carote al supermercato. - You are buying some carrots at the supermarket.

This is the main difference between the two words! Alcuni or alcune must be used with a plural noun, like carote (carrots). Qualche is always followed by a singular noun, even when it means "some" or "a few". To get the exact same meaning, you could also say qualche carota.

Why does alcune end in an "e"?

Because it has to match the gender and number of the noun it describes. Carote is feminine and plural, so we use the feminine plural form alcune. If you were buying masculine plural items, like tomatoes (pomodori), you would use alcuni pomodori.

Why is it al supermercato instead of just a supermercato?

In Italian, when talking about being at or going to the supermarket, you almost always include the definite article (the word for "the"). Al is a combined preposition made by merging a (at/to) and il (the masculine singular "the").

The translation says "You are buying," but isn't compri just the normal present tense?

Yes, it is! The Italian simple present tense covers both "you buy" (a general habit) and "you are buying" (an action happening right now). You don't need a special continuous tense to say what you are doing in this moment.