There is a wide range of effects available from simple foot pedals for one particular sound up to multi-effect rack which you connect together with your amp.
What, how and when to use an effect is a matter of experience and taste, depending on the function the bass has in a particular music piece; groove or solo. Start with a neutral clean sound and experiment with that by changing the tone. E.g. boost the low end (bass boost) and the top end (treble boost) and take the mid-range back to give more sound definition.
The Equalizer is a useful effect to balance your sound out, especially when you have some uneven sounding notes. You can either make them stronger by boosting a particular frequency or to reduce an overpowering sound. Some more sophisticated amps have an inbuilt E.Q.
Reverb is a great effect to create an open space effect and works well in conjunction with most other effects. Be careful not to add too much reverb as it makes the whole sound too distant and loses definition.
Distortion is mostly used by guitar players for lead playing. Never use it when you playa groove! Use it very sparingly only when there is a suitable spot for it – maybe only once in a gig.
The volume pedal is the best effect to use if you are really interested in dynamic playing, because you can come out of nowhere up to full volume and back. It is the same idea as the volume control on your bass, but is controlled by your foot while you play.
The compressor evens out your dynamics. It has almost the opposite effect of a volume pedal and is very useful if you have to play very evenly and steadily especially with slapping or tapping techniques.
The octave divider adds a lower octave to the note you are playing. So when you play one note on your bass guitar, two notes sound. This effect is only useful for the higher notes on the bass guitar as the lower notes will sound too "muddy".
The chorus pedal gives a "swirling stereo" effect on the notes you play. Best suited for solo and melody playing.
The digital delay creates an echo. This echo can be adjusted to be louder or softer than the original note played. It can also be set as a short echo (a reverb effect) or a long echo which repeats the notes over and over.
There are many other effects pedals and new or modified effects appear regularly.