Learn Finnish: Se on tuoli.

Finnish doesn't have words for "a" or "the" (articles). The word tuoli alone can mean "a chair", "the chair", or just "chair", depending on the context. Here, "a chair" makes the most sense in English.

Can se mean anything else besides "it"?

Yes, se can also mean "that". In Finnish, tämä is "this" (close by), tuo is "that" (further away), and se is usually "it" (referring to something already mentioned) or "that" (pointing to something out of sight).

Is on always used for "is"?

Yes, when talking about a third person or a single thing (like he, she, or it). on is the "is" form of the Finnish verb for "to be". You use it directly after se just like in English.

Why is tuoli in its basic dictionary form here?

When you are simply identifying what an object is using the se on... pattern, the noun stays in its basic form (the nominative case). You don't need to add any special endings to it.