cop·u·la [kop-yuh-luh]
In this case especially, です functions as definition 2, and is mostly equivalent to the English "to be" which is itself a verb.
Also, です is the simplified form of であります, yes, あります.
So です is one of three verbs (including ある and いる) which denote existence, though です is generally more of a statement of something's state of being rather than its existence in time/space.
Example:
それはいぬです。
That is a dog.
そこにいぬがいます。
There is a dog.
あそこにいぬがあります。
There is a [stuffed
/dead
] dog over there.
In all three cases the verbs are translated as "is", and tell you something about an object/creature's existence; either what or where it is.
Even though です is often dropped from casual speech, remember that so too can ある and いる. If someone asked you, 「どこにいぬがいる」, you could answer simply, 「あそこ」, or even 「あそこです」 if you wanted to; just as easily as, 「あそこにいる」.
And to reply to Admetus' question, some kanji DO include a small つ as part of one or more of their readings, however, the small つ gained from forming て / た-form is always written in kana, so 行って is how you write the て-form of 行く.