Will someone please explain the technical differences between species and breeds to me again?

Let me explain my puzzlement in simple terms:

There are just over 900 species of bats. Helluva lot of species of bats, almost 1/4 of the counted species of mammals. (There are more species of rodents, however.)

Then there are all those breeds of dogs of which I've been thinking because of something AnnaS referred to somewhere else. And I do understand that breeds are different from species.

However, if a poodle mates a poodle--and they are from a registered line of poodles--the result will be a poodle and not a collie.

And I understand that a poodle and a collie can mate thereby producing another dog that is neither poodle nor collie, but still dog.

And I understand that in oaks, at least, that there can be a cross-species pollination occasionally. One kind of oak can conceivably pollinate another and a strange oak will appear.

But *here comes my question:

Is it that among mammals, not one of those bat species can mate with another and produce a new strange bat as can those occasional cross-species oak pollinations?

Does that mean that one species of tiger cannot breed with another species of tiger? (Seems ridiculous. I would think a Siberian could breed with a Bengal, just as a breed of dog could with another...but I'm no biologist.)

So, will somebody please explain this species/breed controversy to me.

Thank you very much.