What is a Hilbert space?
Hilbert space is a set of elements with certain properties, namely:
it's a vector space (so, there are operations on its elements typical for vectors, like multiplication by a real number and addition that satisfy commutative and associative laws);
there is a scalar (sometimes called inner or dot) product between any two elements that results in a real number.
For example, our three-dimensional Euclidean space is an example of a Hilbert space with scalar product of
More interesting example is a space of all continuous functions on a segment
In quantum physics Hilbert space plays a very important role as a function that describes the state of a system
I can recommend
as an introduction into usage of Hilbert space in quantum physics.