Arpeggios are nothing but chords broken out and played one note at a time So a C7 chord would make a C7 arpeggio which would look like this...C-E-G-Bb
For soloing on guitar (someone correct me if I'm wrong) guitarists are more concerned with scales, which are related to arpeggios....for example the C major triad is formed by stacking 3rds on top of one another C-E-G. If we continue to stack thirds we get C-E-G-B-D-F-A which is a C13 cord, re-arrange it and you have the C major scale C-D-E-F-G-A-B-C. All scales are related to a chord/visa versa and most are related to a few different scales/chords. I now know that a C chord and the C Major scale are related, so I can use the C Major scale to form a line over C chord.
As for how to use an arpeggio on bass...
Any given song has a chord progression, whether its one chord over and over or something more complex. We will use the 12 bar blue for an example. Every bar in the 12 bar blues is represented as a chord, in G it would look something like G7-G7-G7-G7-C7-C7-G7-G7-D7-D7-G7-G7 (there are other variations but this one is simple). We could also write it as a mathematical relationship (I-I-I-I-IV-IV-I-I-V-V-I-I) because the chord progression is formed from the 1,4 and 5 tones in the G major scale.
In the blues every first beat of the measure would be the root note of the chord, but you still have 3 beats to fill before the next root note, most bassists would play a pattern formed out of an arpeggio (sometimes with a few scale tones or "blue" tones from outside the scale thrown in).
So for this blues song I could play a line like this in 8th notes. | G-G-B-B-D-D-F-D | G-G-B-B-D-D-F-D | G-G-B-B-D-D-F-D | G-G-B-B-D-D-F-D | C-C-E-E-G-G-Bb-G | C-C-E-E-G-G-Bb-G | G-G-B-B-D-D-F-D | G-G-B-B-D-D-F-D | Ext...
This whole line uses nothing but chord tones (arpeggios) to form a bass line. A more advanced blues player might start using scale tones to spice up the line, and would also start creating variations and mixing it up, but would still place his emphasis on those chord tones, the root, third, 5th and flat7.
In summery:
Arpeggios are just chords played 1 note at a time
Arpeggios/chords and scales are related and you can form a scale from an arpeggio.
Arpeggios are the key to understanding song structure and give you the strongest set of tones to form your line from.
If you need some more basic info (Ie: I confused the hell out of you) go here
http://studybass.com.
If you need some more advanced information and want to learn more about scale formation ext. then go here
http://teoria.com