Learn German: Ich brauche irgendwelche leeren Kartons für das Zeug. - I need some empty cardboard boxes for the stuff.

irgendein- translates to "any" or "some," but it is strictly used with singular nouns (like "any box" = irgendeinen Karton). Because Kartons is plural, you must use the plural equivalent, which is irgendwelch-. Since it's the direct object in the accusative case, it takes the plural ending -e, becoming irgendwelche.

Why does the adjective leeren end in -en? Shouldn't it match irgendwelche?

This is a tricky part of adjective declension! The word irgendwelche acts like a definite plural article (a "der-word"). When an adjective comes after a definite article in the plural, it takes the weak ending -en in all cases. Therefore, we get irgendwelche leeren Kartons.

What nuance does irgendwelche add compared to just saying "Ich brauche leere Kartons"?

Saying Ich brauche leere Kartons simply means "I need empty boxes." Adding irgendwelche heavily stresses the indifference: you really don't care which boxes they are, what size they are, or where they come from. Absolutely any empty boxes will do.

Why does Kartons take an -s plural instead of -e or -en?