Existence with いる and ある plus である

The verbs いる and ある express existence. The difference between the two is whether the subject is something “animated” or not. An animal and a person take いる, but a table and a sandwich take ある. Something like a car, which moves but isn't alive, can sometimes use one or the other depending on context and focus of the sentence. Don’t be fooled into thinking this is about “living” or not. It is not. For example typhoons take いる because they move on their own. Robots in sci-fi movies also take いる. Sometimes people can take ある as well, especially in old language and set phrases.

家には猫がいる
In the house there is a cat.

店がある
There is a shop.

Sometimes, when people want to state that they have/own something, they use ある or いる. They technically don't mean "to have", but it’s an equivalent translation to how we use “have/possess” in English. It's just a wording difference.

デスクがある
There's a desk / I have a desk

私の家には猫がいる
I have a cat.

いる conjugates as an ichidan verb.

ある conjugates as a godan verb, but its negative conjugation is irregular.

店はない
There is no store.

Now that we know ある, we can bring up the origin of だ: である. である is a formal version of だ (to be). Unlike だ and です, である can be used with relative clauses that attach to nouns:

*友達だ人  
(ungrammatical nonsense)

友達である人
Person who is a friend (formal)

You might remember that we used ではない as the negative of だ. ではない is basically である's negative.

である
であった
で(は)ない

Note: でない also exists, but for some quirk of the language the addition of は has become the standard/expected version when negating である. There is no real reason for it other than it is how it is. You might still see でない especially in more formal written media or certain set phrases.