Distortion and overdrive pedals will give your bass guitar or electric guitar an edgier, harder tone.
An overdrive pedal will add gain to your electric guitar signal, causing it to break up or distort a bit (or more) depending on how much is used.
A distortion pedal produces more gain and clips the signal even harder.
You will find distortion and overdrive pedals from brands such as MXR, Catialinbread, Wampler, Keeley, Electro-Harmonix, and Xotic.
If you are looking for similar sounds, check out a fuzz pedal.