The Moog synthesizer (Moog rhymes with vogue), as you may or may not know, is the "It" synthesizer when it comes to synthesizers. Incredibly versatile, they can either sound kitsch or pulverising.

The Beatles recording Abbey Road. Paul meets a Moog.
Designed by Dr. Robert Moog, the Moog Synthesizer can refer to a number of analogue synthesizers first manufactured by Moog Music in the early 1950s. However, it was not until The Monterey International Pop Festival in 1967, where it was demonstrated, that it started to become more commercially successful. After Monterey, acts such as The Byrds and Simon and Garfunkel started to incorporate the Moog into their work. The Beatles used the Moog Synthesizer extensively on their Abbey Road album. Shortly after recording Abbey Road, George Harrison even produced an entire album using the Moog called Electronic Sound.
For your listening pleasure, I have compiled a list of ten songs that featured the iconic Moog Synthesizer:
1. Strange Days - The Doors, Strange Days (1967)
2. Maxwell's Silver Hammer, Because, Here Comes the Sun and I Want You (She's so Heavy) - The Beatles, Abbey Road (1969)
3. Wish you Were Here - Pink Floyd, Dark Side of the Moon (1973)
4. I Feel Love - Donna Summer, I Remember Yesterday (1977)
5. Whip It - Devo, Freedom of Choice (1980)
6. Rio - Duran Duran, Rio (1982)
7. Blue Monday - New Order, Power, Corruption and Lies (1983)

The Guardian has a great potted history of The Moog. Also here is a pretty definitive list of albums and songs that feature the Moog.
Images from here and here.