A Nakajima variety is an algebraic variety that is associated to a quiver and some extra data. In this talk I will give an overview of the construction of quiver varieties and of moduli spaces of quiver representations. After reviewing some basic results from GIT, I will talk about stability conditions for representations of quivers, then I will use these facts to define quiver varieties and show that in some cases one can arrange things so that the quiver varieties are smooth and symplectic. I will describe in detail the example of the quiver with one vertex and one loop, and show that how this case relates to the Hilbert scheme of points on \mathbb{C}^2.