In the days of homo erectus, people dealt with mathematics by grunting and gesturing and occasionally piling up rocks. One could get an idea of one, few, or many, but could not do terribly sophisticated math or play arcade games.
Eventually, homo sapiens came along. They worked out all sorts of mathematics and ways to represent them. Using them, they were able to make the Pyramids, and the Great Wall of China, and Gothic cathedrals but it was still a long time before they approached the skills necessary to produce machines that could play any sort of game.
In the 19th century Charles Babbage designed a machine called The Difference Engine which would have been a sort of true computer. It might have been able to play chess and "Star Trek", but still couldn't have played an arcade game. Anyway, the technology wasn't quite up to producing it yet.
Electronic computers came about during WWII, to compute shell trajectories. It wasn't a long time before they were playing chess. But they still couldn't do an arcade game, since they lacked really convenient input methods or video displays.
In the Sixties, we finally saw a video game of sorts. It was a table tennis game on an analog computer. It wasn't something for the home, however. It did make it so that when Atari and Magnavox produced video table tennis nether could claim ownership of the idea of a video game.
The Seventies saw the Birth of the Personal Computer. Not that thing that IBM produced and gave the name; no, machines built by crazed hobbyists using the new "microprocessors". Things like the Altair and the Apple. Initially, these were a bit daunting to the regular person; the early Altairs were programmed by a series of toggle switches. Some hobbyist machines used paper tape for mass storage; not so shocking when one remembers that the big machines still used paper punch cards. In this primitive time, people actually typed in BASIC games from books and magazines.
The Seventies also saw the birth of the Video Game, both in the home and the arcade. For a couple of hundred dollars, one could play those arcade game that homo erectus wished he could play. Okay, they weren't terribly colorful or high resolution and the sound was primitive but it was an improvement over grunting.
The Apples and C/PM machines cost as much as our modern computers, with a lot less capability. But that created an opening for cheaper machines. One company to produce a cheap computer was Sinclair Research in Britain. Initially, they produced the ZX80; it had a black and white 256 X 192 display, 1k of RAM, and a 4K ROM, all for under $200 US. It was fairly quickly succeeded by the ZX81; around the same price, but with an 8k ROM and the ability to produce moving graphics. Both machines used ordinary cassette tapes for mass storage. Limited, sure, but a whole lot cheaper than anything else out there and with creative programming one could do a lot. Of course there was eventually a 16k RAM pack.
The ZX81 was a great success, and spawned a wondrous successor; the ZX Spectrum. While it didn't have a higher resolution display, it did have a 16 color display and simple sound. It was about as cheap as it's ancestors, so it was the cheapist color computer ever up to that point. After a while, you could get 48k versions. Timex sold variants of the Sinclair machines in the US. There was very little change to the ZX81, but the American Spectrum (TS2068) added a three channel sound chip, a cartridge port, additional video modes, and joystick ports. Eventually, versions of the Speccy had 128k, a sound chip, and even a disk drive.
The Spectrum sold millions, and there were thousands of games written for it. While not all were grand and wonderful, many were. Sure, they lacked super high color resolutions and CD quality sound and that stuff, but they had quality game play. Plus they were cheap.
Eventually the Spectrum went away, although I hear there's still a clone being produced in Russia. Well, it sort of went away, for it lives on through the magic of emulation. Yup, you can still play those Speccy games on whatever machine you use, as there seems to be a Speccy emulator for every machine that exists. The best one for limited machines like my puny 66Mhz 486 is X128 ; it's an MS-DOS emulator but it works fine under Windows 95. The true Windows emulators are just too slow, and one annoyed me by requiring a 256 color display. X128 emulates the original Speccy, the 128k versions and the two Russian clones, the Scorpion and the Pentagon.
Once you have your emulator, you'll want games unless you really want to recreate the archaic computing experience and type in stuff from 101 BASIC Computer Games. One of the best sites for that is World of ZX Spectrum ; there's a slew of games there, and one can also buy a CD ROM with all the site's games. You should (maybe must) stop by Planet Sinclair ; this site has Sinclair information for all Sinclair's machines and links to a whole lot of Sinclair sites. It also has ZX81, Speccy, and QL (16 bit machine Sinclair produced) emulators for several platforms; some may be better than X128 if your machine is slower or otherwise differs from mine.
Another emulator of interest is the Multi-machine Emulator ; it not only emulates the ZX Spectrum, but the ZX80, the ZX81, the Amstrad CPC, and some more obscure machines like the Jupiter Ace. It requires DirectX 5.0 and was rather slow on my 66 Mhz 486, but works well on my IBM/Cryix P150.
If you're interested in emulating other systems, have a look at ARRGH, the Anally Retentive Retro Games Home. Lots of stuff, reviewed, and an amusing page when you first get there.
But what about the morality of emulation? I've seen suggestions that if you own the original programs it's fine to emulate them, but that's questionable; I understand that Nintendo doesn't accept that argument for their stuff. And if you don't own the original programs you'd be out of luck. Fortunately for Speccy fans Amstrad, the company that bought the Sinclair machines, made the ROMS freeware; you can distribute them as long as you don't charge for them. Most Spectrum emulators are freeware.
As for the programs, well, it depends. Many of the game companies are out of business, so who really owns the programs may be hard to know. And some companies and/or authors put their games into the freeware category; the site Free Games had a list and even links to some sites where the authors have made them available, but that site apparently has gone away alas.
Ah, but what if you don't want to bother with an emulator, but still want to try some of the old game anyway? Well, many of them have been converted to DOS or Windows; go to Remakes for several of them. The site also has some Speccy screen savers and several links to sites of interest.
A site of special interest if you are at all familiar with the Mattel Intellivision is The Blue Sky Rangers. "The Blue Sky Rangers" are the original Intellivision programmers and at their site there's a host of information about the Intellivision and the Aquarius. but most excitingly you can pick up conversions of some of the old Intellivision games. They're great conversions; it's just like playing on the old game console with the exception of the controllers. There's six games in two zip files (and Mac versions), eventually they'll have more and there is now a CD ROM with bunches of programs including some that were never released.
Archaic computing; the way for those who can't or won't run mult-meg games requiring 300 Mhz Pentiums!