Although there are some hi-tech materials in use, the very large majority of bass guitars are made of wood. The critical consideration here is the hardness or softness. It's actually quite common to see a combination of different woods employed in a single instrument to create a balanced overall timbre. Hard woods such as maple, rosewood and ebony give a bright tone with plenty of attack, whereas softer woods such as alder, swamp ash and the coincidentally named basswood, give a more subtle and darker sound.
Most woods have imperfections somewhere in the grain that can produce 'dead spots' - notes with less sustain or resonance - so some companies such as Status and Modulus use synthetic graphite in the construction of their instruments. This gives the bass enhanced sustain, and an even sound across its range.