Hilbert space
A Hilbert space is the abstract arena in which quantum states live. Mathematically it is a vector space — you can add states together and scale them by numbers — equipped with an inner product that lets you ask how much one state overlaps with another, plus a completeness condition that keeps the mathematics well behaved even when the space has infinitely many dimensions. Every possible state of a quantum system is a vector in this space.
It helps to start from ordinary arrows in three-dimensional space, where you can add two arrows, stretch them, and measure angles between them with a dot product. A Hilbert space is the same idea taken much further: the 'vectors' are quantum states, the numbers you scale by are complex rather than just real, and there can be infinitely many independent directions. A single particle's position-state, for instance, needs one direction for every point it could be found at.
Why bother with such an abstract picture? Because it lets the same compact rules describe a spin that has only two states and a particle that can be anywhere along a line. Once you accept that states are vectors and measurements correspond to operators acting on them, the whole machinery of quantum mechanics — superposition, probabilities, evolution in time — becomes geometry in this space, which is far cleaner than juggling wavefunctions and matrices separately.
Every quantum state is a vector in a Hilbert space; measurements and overlaps are read off from its inner product.
A Hilbert space is not physical space. Its 'dimensions' are independent states, not directions you can point to; a particle moving in three-dimensional space already needs an infinite-dimensional Hilbert space to describe it.