This is a technique popularized most notably by Yngwie Malmsteen in the mid eighties, but arpeggios have been around as long as music has. Even the song Ain't Talkin bout Love from Van-Halen uses a form of A minor arpeggio that adds the sixth followed by a G arpeggio with no 3rd. An even earlier example would be the into to Purple Haze from Jimi Hendrix, the main riff uses an E minor 7th arpeggio.

Audio example:

E|---------0-1-----3--------------------------------|
B|-----1-------1-----3------------------------------|
G|-------2-------2-----0----------------------------|
D|---2----------------------------------------------|
A|-0----------------------3-2-/3--------------------|
E|--------------------------------------------------|

 

Example #2 - Purple Haze

Audio example:

E|-------------------------|------------------------|
B|----------8--------------|------------------------|
G|-------7-----------------|------------------------|
D|---/9--------7--------(7)|--5---------------------|
A|-------------------------|-----5-----5/7----------|
E|-------------------------|--------0---------------|

For soloing, arpeggios are a very effective technique to add some flash and speed over a chord progression. The following example simply plays arpeggios that follow the chord. A Minor + A Minor arpeggio, C Major + C Major arpeggio and E Minor + E Minor arpeggio then resolving back to the A Minor + A Minor arpeggio

 

Example #3

               A Minor                           A Minor

E|--------------8-p5----------------|--------------8-p5----------------|
B|------------5------5--------------|------------5------5--------------|
G|----------5----------5------------|----------5----------5------------|
D|--------7--------------7----------|--------7--------------7----------|
A|------7------------------7--------|------7------------------7--------|
E|-5-h8----------------------5------|-5-h8----------------------5------|

                     C Major

E|-------------------8-12-p8-------------------------------------------|
B|---------------------------------------------------------------------|
G|-------------9-h12---------12-p9-------------------------------------|
D|----------10---------------------10----------------------------------|
A|-------10---------------------------10-------------------------------|
E|-8-h12---------------------------------12-p8-------------------------|

                     C Major

E|-------------------8-12-p8-------------------------------------------|
B|---------------------------------------------------------------------|
G|-------------9-h12---------12-p9-------------------------------------|
D|----------10---------------------10----------------------------------|
A|-------10---------------------------10-------------------------------|
E|-8-h12---------------------------------12-p8-------------------------|

               E Minor                           E Minor

E|-------------7-12-p7--------------|-------------7-12-p7--------------|
B|-----------8---------8------------|-----------8---------8------------|
G|---------9-------------9----------|---------9-------------9----------|
D|-------9-----------------9--------|-------9-----------------9--------|
A|-7-h10---------------------10-p7--|-7-h10---------------------10-p7--|
E|----------------------------------|----------------------------------|

 

However, don't get into the rut of only playing A minor arpeggios over A minor chords or E minor arpeggios over E minor chords, etc. This will leave your playing to dull and predictable. Instead try playing a C Major arpeggio or an E Minor arpeggio over an A Minor chord. This will add to the chords sound. In the case of the C Major arpeggio, you are adding the 7th to the A chord. By playing an E Minor over the A Minor the 7th and 9th tones to the chord.

The following audio example plays the same arpeggio patterns while the other guitar stays on the A Minor chord.

Example #4