Yggbert is pretty much right.
The short version is that です means “to be” and いる/ある means “there is/are”.

That is a dog. – それは犬です。
That dog is mine. – その犬は私のです。
The dog’s name is Baxter. – 犬の名前はBaxterです。

There is a dog over there. – 犬があそこにいる。

There can be be combinations of the two as well.
The dog that is over there is mine. – あそこにいる犬は私のです。

Where it gets confusing is the rather haphazard use of です to take the place of other verbs. (similar to the verb “do” in English)
For example, if asked
What do bears eat? – くまは何を食べますか？
You could answer
They eat fish. – 魚をたべます。
But you could also say
魚です。 – Which would translate directly to “It is fish.” or “It’s a fish.”

です can replace いる/ある too.
What is on the desk? – 机の上に何がありますか？
The full answer in English would be
There is a book. – 本があります。
But an answer that is just as good (although in English not grammatically correct) is
A book. – 本です。 Now, the です here does not mean “a” it actually doesn’t mean much of anything but is is kind of a place holder for ある.

Just like in English if you ask
Who likes hamburgers?
You can answer – I do. where “do” replaces “like hamburgers”

Maybe that makes things worse…sorry.