Cyber People

Italo disco from 1984, Cyber People with "Void Vision (Slow Version)":

Tyndall

Nik Tyndall and Rudolf Langer with their 3rd LP, Reflexionen (1982).
"Strassen Der Nacht":

Human League

the Human League from later in their career with their 1986 single, "Human":

Angelo Badalamenti

Angelo Badalamenti recorded the scores for several of David Lynch's films.
"Twin Peaks Theme", as it appears in the opening segment of the pilot episode of David Lynch's Twin Peaks:

"Fred's World" from the Lost Highway soundtrack (1997):

"Mysteries of Love" as it appears in Blue Velvet (1987):

Billy Idol

if the 1980's had just one douche bag to call it's own, it would be Billy Idol. here he is showing his less "hard" side with two of his most popular ballads.
"Eyes Without A Face" from Rebel Yell (1983):

"Sweet Sixteen" from Whiplash Smile (1986):

Unknown DJ

Andre Manuel, the Unknown DJ, early west-coast electro beats.
"Basstronic" single, 1988:

better audio: http://youtu.be/ytddqtWSP-E

Martin Newell

Martin Newell was a strong force for the DIY camp during the new wave days. He produced, recorded, and played (nearly) all of the instrumentation for his solo recordings. He also wrote and played under the name Cleaners from Venus with Lol Elliot in the early 80's, releasing most of their work together from home on cassette. The following are from his first cassette as Martin Newell, Songs for a Fallow Land (1985).
"Julie Profumo":

"Gamma Ray Blue":

Giorgio Moroder

the Godfather of Italian electro, Giorgio Moroder. A promotional TV video for the 1984 Olympics in LA:

R. Stevie Moore

live on NYC public access TV, the Scott & Gary Show in 1984.
"Showbiz is Dead" from Dumb Philosophy (1981):

"I've Begun to Fall in Love":

The Associates

the Associates, led by Billy Mackenzie, were one of the more successful new wave groups of their day; after touring with the Cure in 1980, John Murphy and Mike Dempsey (previously of the Cure) joined the group, and they landed album of the year in 1982 for their third album, Sulk. the following are sessions recorded for John Peel's radio show in April of 1981.
"It's Better This Way":

"Nude Spoons":

"Ulcragyceptimol":

Afrika Bambaataa & the Soulsonic Force

Afrika Bambaataa & the Soulsonic Force were instrumental in the development of hip-hop (Afrika actually coined the term) and electro-funk with their unique blend of European electronic music, american funk, and rap. They helped push hip hop to the forefront of American dance music, and were one of the first hip hop groups to carry strong (and positive) political and social undertones.
"Looking for the Perfect Beat" video for the 1983 single:

"Planet Rock" from the 1982 Planet Patrol/Soul Sonic Force split EP:

"Renegades of Funk" video for the 1983 single:

Angelo & His Egos

obscure Greek new wave, Angelo & His Egos released one record, Egotrip (1982):
"Blackness":

The Short Wave Mystery

The only album ever released by the group was a four song EP of sorts called Pilots which was recorded while the members were in community college in California. 1000 copies were pressed, 500 of which were not even sold. the record, when for sale, can fetch up to $1000.
"Nice Girl":

"Ecaterina Szabo":

Tomo Akikawabaya

Japanese mystery man Tomo Akikawabaya made his brief appearance in the mid 80's with several self-released singles, each featuring model Rena Anju on the cover, and promptly fell off the face of the Earth. there are no pictures of this man known to exist, he gave no interviews, never publicly performed, and he has (presumably) not released any more music since 1986. All instruments and noises played by Tomo Akikawabaya.
"Diamond" from the 1985 single:

Side B - "Dream of No Pillow":

Aloa

Aloa were a German synth duo, releasing only one LP in the early 80's. The LP has become a collector's item, selling for quite a lot of money when it can be found.
"Flamingo"

"Weisser Wal"

That's Hot

someone's got a case of the mondays... me. This is a very self-indulgent post.
Early video art by G.G. Aries, music by Larry Gibbs, from California Images: Hi-Fi For the Eyes, 1983:

Soft Cell

British synthpop duo best known for their hit single, "Tainted Love," Soft Cell with "Sex Dwarf" from Non-Stop Erotic Cabaret (1981). The song also had a banned video in the UK.

Lizzy Mercier Descloux

a French progenitor(tress?) of early no wave (writing as Rosa Yemen) and later various new wave, disco, and worldbeat sounds, Lizzy Mercier Descloux certainly did not stick to one style of songwriting. The following song, from her second album (as Lizzy), Mambo Nassau (1981) perhaps best melded her early no-wave and new wave roots with her danceable beats and definitely funk-ified world music consciousness:

Jackie Quartz

"Émotion (Le Jour Se Lève)" single, 1987:

Trisomie 21

Trisomie 21 has been a highly regarded staple in French electronic music since the early 80's up until the present day. "Memories" from Chapter IV (1986):