Learn Finnish: Minä toin uuden kattilan.

The basic verb is tuoda (to bring), which is a Type 2 verb ending in -da. To make the past tense, we take the stem tuo-, drop the first vowel of the double vowel sound (the u), add the past tense marker -i-, and attach the personal ending -n for 'I'. This turns tuo- into toin.

Why are the words uusi (new) and kattila (pot) written as uuden kattilan?

They make up the object of the sentence. Because bringing the pot is a fully completed action and you are bringing one whole pot, the object takes the accusative case. In the singular, this case takes an -n ending (looking exactly like the genitive case). The adjective must match the noun, so both get the ending.

Why does uusi change so drastically to uuden instead of just uusin?

Finnish words ending in -si have a special behavior. When you need to add grammatical endings to them, the -si part usually changes to -te- or -de- to form the stem. For the word uusi, the stem becomes uude- before we add the -n ending.

Can I leave out the word minä?