Formed by Lemmy (Ian Kilmister) in 1975 (!) after he was kicked out of Hawkwind, the new band (named after the last song that Lemmy wrote for Hawkwind) had a tough start. In fact, by 1977, the trio including guitarist "Fast" Eddie Clarke and drummer "Philthy Animal" (Phil Taylor) were flat broke. The band decided to do a farewell show before breaking up, which led to a one album contract with Chiswick Records. The following year, they managed to sign a one single deal with Bronze Records, which was extended several times as the band's popularity grew in the U.K.
"Ace of Spades," from the 1980 album of the same name would go on to become the band's signature song. The video is a simple performance piece and it fits well with the group's dark, rocking sound. Motörhead is definitely a "let the music do the talking" sort of group, and it totally comes off in the video. I also think that their sound has held up remarkably well (this is a 30 year old song!).
Lemmy continued to record and perform with Motörhead until his death in Deceber 2015. However, "Fast" Eddie Clarke left the group in 1982 to form Fastway, though he has played with Lemmy several times during the past few years. "Philthy Animal" (Phil Taylor) left Motörhead in 1984, rejoined the group in 1987 and left for good in 1992.
Motörhead recorded a cool, acoustic version of "Ace of Spades" for a Kronenbourg beer commercial in 2010; this version (well worth a listen) is below.