Even if you have access to an electronic tuner, tuning a bass guitar by ear is a very important skill for a bassist to learn. This page should help you understand what you are listening for and how to do it.
Pitch and Frequency
Without getting into too many scientific details about sound and pitch, there are a few basic things to understand. How high or how low a pitch is depends on its frequency. Frequency is how frequently something (like a bass string) vibrates. The faster, or more frequently, something vibrates, the higher its pitch.
In Tune and Out-of-Tune
When you are tuning your bass strings, you are trying to match the frequency, or vibration pattern, of one note to another. If two notes are out-of-tune with one another, the vibration patterns clash. You should hear a wah-wah-wah sound from the two notes clashing.
For example, listen to these two out-of-tune bass notes (99kB).
When two notes are in tune, the vibration patterns match. This is what you're trying to acheive when you tune your bass strings.
The next page will show you what tuning to a note sounds like and the process of doing it.