Nintendo was founded in 1889 by Fusajiro Yamauchi, grandfather of Hiroshi Yamauchi (see pic), who was Nintendo's president from 1949 to June 2002. For the first 70 years of its existance, Nintendo manufactured japanese playing-cards; in 1963 they began to make toys to enlarge their range of products. Then, from 1970 on, they started to produce electronic devices for theme parks, and in 1977, in coorporation with Mitsubishi Electric, they built their first 'at-home' videogames, as well as arcade machines. In 1980, when the videogame-market was at full bloom, Nintendo decided to expand to the USA - Nintendo of America Inc. was founded; their first success in the USA was the 'Game & Watch' LCD-games series. In 1981, Nintendo introduced Donkey Kong as an arcade-machine, probably their most successful videogame ever, and its hero Mario became a symbol for the company.
Although in 1983, the videogame-market was nearly dead in the U.S. (due to the infamous 'videogame shakeout'), Nintendo introduced their Family Computer System (called 'Famicom' by its fans) in Japan, followed by Donkey Kong jr., Mario Bros. and the VS.System, which allowed players to directly compete to each other.
Pushed by the Famicom's success in Japan, Nintendo dared to launch it in the U.S. in 1986 - although analysts prophesized a financial fiasco. But they were wrong! The so-called 'Nintendo Entertainment System' reincarnated the dead videogame-market; Nintendo sold over 1 million pieces in the first year!
Encouraged by Nintendo's success, the market began to regenerate, and ATARI and Sega also introduced new videogame machines (ATARI the VCS 7800 and Sega the Master System). In 1987, the NES was introduced in Europe and Nintendo began to fully concentrate on the private consumer.
Another milestone in Nintendo-history is the 'Super Famicom' (1990, called 'Super Nintendo Entertainment System' in Europe and the U.S.), that rivaled the Sega Mega Drive and the homecomputers these days. Then, in 1995, they built the amazing Nintendo 64 (formerly known as 'Ultra 64'), which was developed in coorporation with Silicon Graphics and MIPS.
Even today, Nintendo successfully creates and sells first-class videogame consoles and handhelds - amazing, if you consider the company's age. But as mentioned earlier, it is no longer led by a Yamauchi - on May 31, 2002, 42-year old Satoru Iwata took over Mr.Yamauchi's seat, who will stay in the company as an advisor (may he live long!).

Also see the Nintendo of America website for more info on actual products.