Unfortunately Quake is an ancient DOS game, and the version of Shareware Quake on ID Software's website, as well as many full Quake CDs, cannot be installed on most modern computers.
The steps in this article should get you to a stage where at least you have DOS Quake installed and running OK. From there you should visit QuakeOne.com and look through their Quake Help pages for more great tips, as well as getting a proper Quake executable that will run on Windows.
I used the version of Shareware Quake on ID's site to verify the steps here, and everything was done on a standard Windows 7 setup. There might be subtle differences if you have a full Quake CD, but they should be obvious and you should not run into any trouble.
Step 1 - what you need
You'll need Quake obviously. If you don't have it you should grab Shareware Quake from ID's Website. It's a hellish slow download but at least you can be certain that what you're getting is the Real Thing. quake106.zip is the one you need.
You should also be able to find a copy of the full CD on eBay, down your local bargain bin, or wherever. You might even get lucky and get a more recent Windows Quake CD, in which case these instructions don't apply to you. It's also available on Steam, so if you go that route you can also ignore these instructions.
Beware of pirated downloads. The guys at ID put a lot of time and effort into making this game, and they deserve some payback. Just don't do it, it's not cool.
Secondly you'll need a copy of DOS Box. It's free, it's small, it's a fast download, and it will also come in handy if you have (or plan to get) any other old DOS games or applications. You're on your own with those though, I only do Quake.
I used the Win32 version of DOS Box 0.73; other versions might look a little different but the steps should mostly be the same.
When I installed DOS Box I got a "this program might not have installed correctly" message, but it worked fine despite that.
Step 2 - preparing
Let's not make this difficult by trying to do fancy stuff. Just unzip that quake106.zip file to C:\Setup (create it if you don't have it) and be done with it. We're in DOS land here so no long names, no spaces, no special characters. If you have a full CD you can alternatively copy the contents of it to C:\Setup.
If you've done it right you should now have 4 files (for Shareware, full may be different) in C:\Setup - deice, install, resource.1 and resource.dat; if they're not there go back and do it again until you've got it right.
Remember, now is not the time to try anything fancy, so don't second-guess things. This is just temporary storage for the purposes of setting things up, so you can move it to where you want to afterwards.
Run DOS Box. You don't have access to your computer's C drive from it, so type "mount c c:\setup" (without the quotes) to get it. Then type "C:" (again without quotes).
Step 3 - installing
Type "install" and press return. You should see the following screen:
Just press the "C" key to select your drive, and then you'll get the next screen:
Now is not the time for anything fancy. QUAKE_SW (or QUAKE for a full CD) is plenty good enough for the purposes of just getting it installed. If you want it somewhere else you can move it later. Press return and the installer will do it's magic.
After a short while the screen will start filling with dots. This is perfectly normal and just means it's working right. When it's done (and it will seem to take quite a while) you'll get the fabled "Prepare to enter Quake!" message, so press any key to see what state of the art technology looked like in 1996.
You can play a bit if you want, but eventually you'll want to get out (using either the menu or the "quit" command), then type "exit" (without quotes) to also get out of DOS Box.
The next steps
In C:\Setup\QUAKE_SW (or C:\Setup\QUAKE) you'll find a folder called ID1. This contains the Quake data files, and is all that you need. Burn a copy of it to CD so that you'll always have it, and store it with your Quake CD (if you have one) so that you'll always be able to find it.
Quake, being ancient, can be easily moved around to anywhere on your PC. Create a folder called Quake on your C drive (or anywhere else you want it) and copy the ID1 folder into there.
You can also delete the Setup folder after this.
Then like I said, head for QuakeOne.com and grab a more up to date Quake engine. This goes in C:\Quake (or whereever you put it), NOT in ID1, and NOT anywhere else on your PC.