Particles: は vs. が vs. を

[The Summoner of Leviathan]

は vs が

は is traditionally called the "topic marker", meaning it marks the topic of the sentence. Easy enough. It can also be used in contrast but that's a tad more complex grammar. It also indicates that the topic is known to both parties, that is why in sentences such as 「猫がこの部屋にあるか？」 you use が because the speaker does not know if there is a cat in the room. You do not use は.

が is traditionally called the "subject marker" though some people find that an inapt name. Grammatically speaking, yes が does mark the subject of the sentence, but it also marks an unknown. Thus question words are always followed by が, not は.

And yes, が does show more emphasis than は. So sometimes it is used instead.

を vs が

This is rather easy.

を is used to mark the direct object of a transitive verb.

が is used to mark the indirect object of an intransitive verb, thus why it is used with the potential form.

The particles have other uses too but I assumed that this was the context we're talking about.

Oh, and particles have pecking orders, so when the object and the topic are the same, は is used. Only で　and に can coexist with は, such as では and には.