Learn Spanish (Latin America): Ellos piden la cuenta y pagan la propina.

Piden comes from the verb pedir (to ask for). It is a stem-changing verb. This means that in the present tense, the e in the root of the word changes to an i for most forms, including the ellos (they) form. That is why it becomes piden instead of "peden".

The English translation says "ask for," but I don't see a word for "for" in the Spanish sentence. Why is that?

The Spanish verb pedir means "to ask for" or "to request" all by itself. You don't need to add an extra word for "for" (like por or para). You just put the thing you want right after the verb, like piden la cuenta (they ask for the bill).

Is pagan also a stem-changing verb like piden?

No, pagan comes from the verb pagar (to pay), which is a completely regular verb in the present tense. Because it is an -ar verb, you just drop the -ar and add the "they" ending (-an) to match ellos, giving you pagan.

What exactly do la cuenta and la propina mean?