Learn Finnish: Koira on ruskea.

Finnish doesn't use articles like "a", "an", or "the". The word koira can mean "a dog" or "the dog" depending on what makes sense in the conversation.

Does the -a at the end of ruskea mean it is in the partitive case?

No, ruskea is just the basic dictionary form of the word "brown". Because a dog is a single, countable object, the describing word stays in its basic form. The partitive form of this word would actually be ruskeaa.

Why doesn't this sentence use the partitive case for the color, like in Maito on valkoista (The milk is white)?

You only use the partitive case for adjectives when you are describing uncountable things, like milk, water, or coffee. Since a dog (koira) is a countable thing, the adjective stays in the basic dictionary form (the nominative case).

Which basic verb does on come from?

On comes from the verb olla, which means "to be". On is the form used for "he is", "she is", or "it is". Since a dog is an "it" (or "he/she"), we use on.