In the good old days, back when the Internet had about 10 users and the web was a far off vision, PCs were called Home computers and anyone could learn to write computer games. Three of us set up in business to write and sell our own games for the ZX Spectrum (Aka Timex / Sinclair) and Commodore 64. We did not sell many games but we were saved by getting into the game conversion business. A publisher would release a hit on the Commodore 64 and we'd write the Timex, Amstrad, MSX, CBM 16 etc versions. It took about 3 months for one person to produce a game.
That was 20 years ago and things have changed a lot since then. The games industry is now a massive multi billion dollar business and development teams can number 50 or more. My first game had me as programmer, artist, and beep designer – (well it was not proper sound!). I did everything. Back then games were written in assembler and you had to learn 6502 and Z80. Nowadays with a few exceptions games are written in C and C ++. The days of back room programmers are not entirely gone- there is also 'retro remake' and 'indie' scene, eg http://retrospec.sgn.net/ but unless you have an exceptionally brilliant idea, you are not likely to get into the mainstream game business as a man man outfit writing Blockbuster PC games.
It's not all bad news though- if you can program to a very good standard and have taught some of the techniques- eg 3d maths, artificial intelligence (ai), and can produce demos of your code then you may be able to get a job in the market. There are plenty of books and websites that will guide you through the intricacies of DirectX and before long you'll be a guru on vertex shaders etc. Sign up on games development sites like http://gamasutra.com/ (it's free) and look for jobs. You will have to be good to get a job and prepared to work long, in some case very long hours. It is quite a tough business – when a game is under development, you'll live breathe and eat it for 18 months-two years. And you probably will not work 9-5 all the time.
There are however other ways to get into the games business and I do not mean console development. There are other 'genres' of games that are much easier to get into.
1. MUD – Multi-user dungeons. These have been around for 30 years and were the precursor to the MMORPG (Massive Multiplayer Online Role-playing Games) that now exist. These are still popular and you can find the source code to several of these. I do not know if any make money- most people are now used to playing them for free. Look here: http://www.ability.org.uk/mud.html
2. Internet / Web based games. There are hundreds of these around. A good starting point is the vast directory at http://www.mpogd.com/ . These range from free to monthly subscriptions. Anyone can set up their own website and games using any web technology- eg Perl, PHP, Java.
3. Play By Mail. This has now largely but not entirely moved into the Internet / Web Games area. Flagship magazine is a good source of information – their website is http://flagshipmagazine.com/
5. Mobile Phone and PDA games. This has become a big growth area recently and is likely to continue its growth as phones get more powerful. Its also one of the easiest to get into though not all phones are the same. All you need to know is how to write Java games running on the J2ME (Java 2 Mobile Edition) platform. Unlike the world, Software Development Kits (SDKS) are available for free from mobile phone manufacturers. Eg http://www.forum.nokia.com/main.html and the best IDEs for J2ME development (Netbeans and Eclipse) are available free. Expect to see MMORPG (Mobile Multiplayer Online Role Play Games) before too long.
Much of this can be done part time in your evenings / weekends but be careful that you do not let it take over your life and also that your employer does not have any claim on software that you develop. Some employers have clauses that give them claim anything you develop. There are ways round this, often involving a wife or loved one who can be assigned ownership but take legal advice if in any doubt.
If you are learning to program games, do not neglect the power of the web. Sourceforge.net is home to thousands of open source projects and many of those are games. Also game source for commercial games Quake I, II and III has been released by ID and can be found here http://www.idsoftware.com/business/techdownloads/
These are GPL'd so you would not be able to write games based on them and sell them without giving away the source code but you could write Mods. Several game modders have released very popular Mods free and have managed to get into the industry because of their quality. There are many games projects where you can sign up and program- this is an excellent way to learn new skills and make friends.