I've got a breaker that has two switches and each switch has a marking of 15A. Does that mean it's a 30A breaker or it's a 15A breaker?

In general those are duplex breakers designed to fit two breakers in a single breaker space. And, as far as I know it is not a 30 amp breaker, it would say 30amps if it were.

If the switches on the breaker are connected together, you have a 15 amp breaker 240VAC. If the switches are separate, you have a 15 amp breaker controlling 2 120VAC circuits. Be careful with this breaker because some people have been known to use a double 120 in place of a 240 breaker. Do you know what is controlled by this breaker?

I've got a breaker that has two switches and each switch has a marking of 15A. Does that mean it's a 30A breaker or it's a 15A breaker?
Regardless if it is a skinny/twin/tandem or a two-pole, it is a 15A breaker.

If it's a 15A breaker then why do the poles each provide power to different items?

If it's a 15A breaker then why do the poles each provide power to different items?
I really don't understand what you're asking, but the best way I can explain it is the breaker is providing 15A to each pole, whether it is a single pole or double pole or twin single pole.
It is either providing:
-15A to one pole; 15A/120V
-15A to two poles; 15A/240V
-or 15A to two separate 120V poles. 2 times 15A/120V

In NO case do you combine the amperages.

So you are saying that my 2 pole breaker is actually supplying 15A to each pole, correct?

So you are saying that my 2 pole breaker is actually supplying 15A to each pole, correct?
Exactly.