Question

What is a Hilbert space?

Answer 1

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 `x=(x_1,x_2,x_3)` and `y=(y_1,y_2,y_3)` equal to `(x,y)=x_1*y_1+x_2*y_2+x_3*y_3`.

More interesting example is a space of all continuous functions on a segment `[a,b]` with a scalar product defined as
`(f,g) = int_a^b[f(x)*g(x)]dx`

In quantum physics Hilbert space plays a very important role as a function that describes the state of a system `Psi` is an element of a Hilbert space.

I can recommend
http://www.phy.ohiou.edu/~elster/lectures/qm1_1p2.pdf
as an introduction into usage of Hilbert space in quantum physics.