Learn Portuguese (Portugal): Quando há pó, eles aspiram-no.

This happens because of the nasal sound at the end of aspiram. Whenever a verb ends in a nasal sound (like -m, -ão, or -õe) and is followed by the direct object pronouns o, a, os, or as, we add an n to the pronoun to make it flow better. So, o becomes -no.

What exactly is the -no replacing in this sentence?

It replaces the word pó (dust). Since pó is a masculine singular noun, we need the masculine singular pronoun o. We then apply the nasal rule from this lesson to change it to -no.

I remember há means there is, but does it change if there are multiple things?

No, it does not. Há comes from the verb haver. Unlike English, which switches between there is and there are, há never changes form for plurals. You use it for both there is dust (há pó) and there are books (há livros).

Does aspirar only mean to vacuum?

In this context, yes, it means to clean with a vacuum cleaner (um aspirador). However, its broader meaning is to suck or to inhale, so it is also the verb you would use for breathing in air.