An arpeggio is like a cross between a chord and a scale. You could consider it simply a scale made from the notes of a chord. When you strum a chord on the guitar, the notes of the chord are all played at once. Although you could argue that there is a slight delay between the notes because of the time it takes the pick to cross all the strings, they are essentially played at the same time.
If you play the chord notes one at a time, i.e., let's say pick down the strings one at a time (like you would in house of the rising sun) then you are playing the chord as an arpeggio, or "arpeggiating" the chord.
An arpeggio doesn't necessarily have to be played while you are holding a chord down with the fretting hand. In fact, arpeggios are probably more often played like a small scale. For example, if you played a C major scale you would be playing the notes C D E F G A B. This can be played all on one string or spread across more than one octave using all the strings. If you were to play a Cmaj7 arpeggio then the notes would be C E G B. You could play this on one string or across all six strings just like you would the C major scale, albeit, with a few less notes.
Arpeggios are useful for making melodic solos and improvising. If you master some licks and melodic phrases for arpeggios then you can play them over any chord sequence and always have something that sounds right and matches the music. Of course if you overdo it, it may end up sounding a bit like elevator music but by incorporating arpeggios with your normal repertoire of scales and licks, you'll find your guitar solos become more creative and professional sounding.