Learn Finnish: Tämä on kissa.

Finnish does not have articles like 'a', 'an', or 'the'. The word kissa can mean 'cat', 'a cat', or 'the cat' depending on the context. In this sentence, it translates best to 'a cat'.

When do I use tämä instead of tuo or se?

You use tämä (this) when the object is physically close to you, the speaker. If the cat were further away, you would use tuo (that). Se (it) is usually used when the animal is out of sight or has already been mentioned.

Can I change the word order and say Kissa on tämä?

Technically yes, but it changes the focus. Tämä on kissa simply states what the object is ('This is a cat'). Kissa on tämä means 'The cat is this one', which you would say if you were pointing out which specific animal is the cat.

I remember the dictionary form for 'to be' is olla. Why are we using on here?

On is the specific form of olla used for 'he', 'she', or 'it' (the third-person singular). Because tämä acts as a singular 'it' in this sentence, we use on, exactly like how you use 'is' instead of 'be' in English.