Learn Finnish: Vedessä on pieni kala. - There is a small fish in the water.

Finnish does not have a dummy word for there. To say that something exists in a location, Finnish uses an existential sentence structure. It puts the location first (vedessä), then the verb (on), and finally the thing that is there (pieni kala).

How do we get vedessä from the basic word vesi (water)?

This involves two common spelling rules. First, old Finnish words ending in -si change their stem to -te- before endings are added, making vete-. Second, adding the -ssä ending closes the syllable and triggers consonant gradation, which weakens the t into a d. That is why vesi plus the -ssä ending becomes vedessä.

I learned that sentences stating what exists somewhere often use the partitive case. Why is kala in the basic nominative form?

You are right that these sentences often use the partitive, but only for plural things or uncountable mass nouns like coffee or water. Because kala is a single, countable object here, it stays in the basic nominative form.

Why does the adjective pieni not take any endings here?