The word arpeggio (ar-peh-jee-oh) is Italian for "like a harp". It is a common technique for playing chords on the harp. To play an arpeggiated chord on the guitar, pick each note of the chord slowly, one string at a time. You can play arpeggios with a plectrum or fingerstyle.
Below is a simple arpeggio study using these chords:
Sweep picking is a more specialized technique, occurring most often in metal. It involves playing a fast arpeggio with a special technique: when switching from one string to the next, mute the note currently ringing by lifting the fretting finger. A sweep can become a rake if notes are muted incorrectly. Rakes can sound nice, but they are not sweeps. Remember only 1 note can ring out at a time or it won't sound good. It takes practice and it helps to start slow and build up speed.
Below is example tablature of sweep picking:
This is not the only way to notate sweeps. Small sweeps can be indicated with grace notes or even the arpeggio notation with the word "sweep" (or, less correctly, "rake") written above.
In a more classical approach, arpeggios must follow a distinct pattern of notes depending on the chord/scale we're playing. This is similar to playing chords note-by-note on a piano (not on a guitar).
The basic chords (the major and minor triads) are composed of three tones: the first, the third and the fifth note of the scale (major or minor, depending on the chord type).
For instance, the C major scale is: C D E F G A B. So, according to the 1-3-5 principle, the C major triad consists of C, E and G. Note that the C major chord on a guitar also consists only of these three notes but they are not always in the 1-3-5 order. Now, while playing "classical arpeggios", you would not just pick around the chord randomly but you would play C, E, G, then C, E, G an octave higher, etc. This is what is called an arpeggio scale. You can play around it, up and down with complete freedom or just use the 1-3-5 pattern as a bass line. This method can also be used with more complex chords (sus4, maj7, etc.) but then it follows a pattern different from 1-3-5 structure, depending on the chord type. In all, this is a very simple but effective method for composing.
While playing guitar, this might not appear as interesting as picking "full" six-string chords but it can be used to give your music a classical edge. It also has a more lead quality to it than using full chords and requires more skill. Playing fast arpeggios like these is sometimes used in metal music with very satisfactory results. The "classical arpeggios" are in no way better than the "harp like chords" and it is ultimately up to the player/composer to choose what is best for the song in question.