Define harmony. The basic building block of music is the note. Groups of notes can be played at the same time, forming chords, and when chords are played successively, they form chord progressions. When we play individual pitches successively, we form melodies. Harmony is the theory behind all of these musical relationships.
Understand chords. In Western music, most harmony is based on chords. Chords are groups of notes built on major or minor triads. In traditional triadic (three-note) harmony, there are always at least three notes in a chord--there can be more than three--but some of the notes may be left out and only "implied" by the harmony.
Get familiar with the concept of chord progressions. A series of chords played one after another is a chord progression. Musicians may describe a specific chord progression ("two measures of C minor, a measure of F seventh and a measure of B-flat," or just "C minor, F seventh, B-flat") or speak more generally of types of chord progressions and keys (a "ii-V-I chord progression in B-flat").
Play the roots. The most important duty of every bass player is to play the root--the lowest note--of each chord. The more you understand harmony, the more you'll see how the bass notes define each chord; play a different note, and everything changes. The other instruments depend on the bass player to outline the harmony by playing the bottom note of each chord.
Connect the chords. Beyond defining the root, bassists are usually asked to create interesting connections between the roots of each chord in the chord progression. Building a walking bass line, for example, consists of outlining the notes in each chord and finding ways to express the chord progression without simply spelling out the chord notes in a mechanical fashion.
Know your fretboard. Expressing harmony is a matter of choosing notes that reinforce or compliment the chords you're accompanying, so it's crucial that you know where the notes are on your fretboard. Many bass instructors teach their students how to look at the fretboard and understand the harmonic possibilities under their fingers, using their ears and their hands to express the harmony at every moment.
Learn how to play chords on bass. It's safe to say that most bass players will not be called upon to play three- and four-note chords in most contexts, but having even a basic idea of how to form chord shapes can help you understand chords, chord progressions and the options available to you at any given moment.