Learn Spanish (Spain): El chico pesó el sobre. - The boy weighed the envelope.

It does, but only if there is no accent mark! Peso (no accent) is the present tense meaning 'I weigh'. However, pesó (with an accent) is the past tense (Pretérito Indefinido) for he, she, or you (formal). Here, it means 'he weighed'.

I thought sobre meant 'on' or 'about'. Why is it translated as 'envelope'?

Sobre is a word with multiple meanings. You are right that it frequently acts as a preposition meaning 'on' or 'about'. But it is also a masculine noun, el sobre, which means 'the envelope'. You can tell it is a noun here because it has the article el right in front of it.

Does the accent mark on pesó change how it is pronounced?

Yes, it tells you to put the spoken emphasis on that final vowel: pe-SÓ. This is really important because if you stress the first syllable instead (PE-so), Spanish speakers will hear the present tense 'I weigh' or the noun for weight, el peso.

This lesson teaches past-tense time markers like ayer (yesterday), but this sentence doesn't have one. Is that okay?

Absolutely. While time markers like anoche (last night) or ayer are great clues that a finished past action happened, you don't strictly need them. The verb ending -ó in pesó is enough on its own to tell us the action is completely finished and in the past.