Learn Croatian: Konačno smo dogovorili popust. - We finally agreed on the discount.

Dogovoriti is the perfective aspect, meaning the action of agreeing was successfully finished. If you used the imperfective form dogovarati, it would mean you were in the process of negotiating or trying to agree, like 'we were discussing a discount.' The stem change from -ovar- to -ovor- is a common way to show this aspect difference.

Why is smo placed right after Konačno?

Smo is a clitic (a 'weak' word without its own stress), and Croatian grammar strictly requires clitics to be in the second position of a sentence. Since Konačno is the first word, smo must immediately follow it.

What case is popust in, and why didn't its ending change?

Popust is the direct object of the verb, so it is in the accusative case. Because it is a masculine non-living (inanimate) noun, its accusative form is exactly the same as its nominative (dictionary) form.

Does dogovorili have anything to do with the verb govoriti (to speak)?

Yes! The root verb is govoriti (to speak). Adding the prefix do- changes the meaning to reaching an end point through speaking, which translates perfectly to 'agreeing' or 'arranging.'