The traditional approach to graphically representing these functions is to draw them in three dimensions with the radial distance at any given θ and φ proportional to |Y|² (see Eric Weisstein's discussion for examples). But I find this kind of representation difficult to understand as a first exposure. The spherical harmonics have nothing to do with the r coordinate. They are functions of θ and φ only. Thus, I like to represent them as color gradients on the surface of a sphere. Below is such a graphical representation. Note that only the real component is being plotted here. The imaginary component would be similar, but rotated about the z axis. Click on the buttons to view the corresponding spherical harmonic.