A Hilbert space is a vector space with an inner product that is used to define a distance function for which it is a complete metric space. That is, the “length” of a vector in the space is given by \(\parallel\vb{v}\parallel = \sqrt{(\vb{v},\vb{v})}\) and the distance between two vectors is defined by \(\parallel \vb{v-w} \parallel = \sqrt{(\vb{v-w}, \vb{v-w})}\). (See here for the definition of inner product for vector spaces over the field of complex numbers.)
The meaning of a complete metric space has to do with with limiting sequences of vectors: if a series of vectors, \(\displaystyle \sum_{k=0}^\infty \vb{v}_k\) converges absolutely — meaning that \(\displaystyle\sum_{k=0}^\infty \parallel \vb{v}_k \parallel < \infty\) — then the partial sums converge to an element in the Hilbert space.
What is an example of a Hilbert space? Suppose that \(H\) is the space of all square-integrable one-dimensional real functions of a real variable. That is, the elements \(f\) of the space \(H\) satisfy \begin{equation}\label{eq:square-integrable} \int_{-\infty}^{\infty} [f(x)]^2 \dd{x} < \infty \end{equation} These functions must die away sufficiently rapidly as \(|x| \to \infty\) for the integral to exist, and this property ensures that the inner product of two such functions remains a function in the Hilbert space. This means that linear combinations of these vectors in the space remain square-integrable and form (normalizable) vectors in the space.