Learn Finnish: Me emme pese ikkunaa tänään. - We do not wash the window today.

Because this is a negative sentence (emme pese). In Finnish, the object of a negative action must always be in the partitive case. The base word is ikkuna, and adding an extra -a makes it ikkunaa.

Why is the main verb just pese instead of pesemme (we wash)?

In negative sentences, the negative verb (emme) already tells us who is doing the action. Because of this, the main verb drops its personal ending (-mme) and just uses its negative base form, which is pese.

Could I drop the word me and just say Emme pese ikkunaa tänään?

Yes! Because the negative verb emme explicitly means "we don't", the pronoun me is optional. Including it just adds a little emphasis to the sentence.

Is the ending on tänään (today) also a partitive ending like on ikkunaa?

No, tänään is simply the standard dictionary word for "today". While it historically comes from an old grammar rule, for modern Finnish learners it is just a fixed vocabulary word you do not need to change.