Learn Finnish: Tämä on kirja.

Finnish does not use articles like 'a', 'an', or 'the'. The word kirja can mean 'book', 'a book', or 'the book' depending on the context. You simply say 'This is book'.

Does tämä specifically mean 'this' as in something right next to me?

Yes! Tämä is used for things that are physically close to the speaker, just like 'this' in English. In this lesson, you are also learning to contrast it with tuo (that, over there) and se (it or that, out of sight).

What is the base form of the verb on?

The dictionary form is olla, which means 'to be'. The word on is the form used for 'he is', 'she is', or 'it is'. Since tämä acts like an 'it' in this sentence, we use on.

If I want to say 'This book is...', do I still use exactly these same words?

You would use the same words, but the structure changes. Tämä on kirja is a complete sentence meaning 'This is a book.' If you want to say 'This book is [good/heavy]', you group the noun and pronoun together: Tämä kirja on...