What is a groupoid? I'm reading a book about gyrovectors and in the first chapter it starts defining something which it calls it a groupoid but doesn't inform the reader about its axioms. so I felt uncomfortable to face a new mathematical definition without knowing the axioms that it should satisfy.
Here is what the book says:

What axioms should a groupoid satisfy? and what's the difference between a groupoid and group? because the name apparently has been derived from the word group.

A groupoid doesn't have any axioms. It's just any set with any binary operation.
So [itex](\mathbb{N},+)[/itex], [itex](\mathbb{N},\cdot)[/itex], [itex](\mathbb{Z},\cdot)[/itex], etc are all groupoids.

The only requirement is that the operation must really be an operation. For example [itex](\mathbb{R},/)[/itex] isn't a groupoid as 1/0 doesn't exist.

A more conventional name for a groupoid is a magma: http://en.wikipedia.org/wiki/Magma_(algebra)
The name magma is preferred, because groupoid often has another meaning in category theory and Lie algebras.

Yes, the operation must be well-defined and the set must be closed under the operation. (for example [itex](\mathbb{N},-)[/itex] is not a groupoid since 2-3 is not a natural number. That are the only requirements.