Mike's PC Tips and Tricks (my opinions -- your mileage may vary)
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Critically important tips and tricks for all computer users

My current (1/22/16) security software recommendations.
For non-techies, important computer: Webroot SecureAnywhere Internet Security Complete ($79.95 list, about $50 online at Walmart for the downloadable version in October 2016; protects 5 PCs for one year).

For non-techies, kid's computer (i.e. no credit card numbers, tax returns, not the end of the world if get an infection and have to reinstall Windows): Microsoft Security Essentials Free!

For techies, I suggest best-of-breed combinations.  See below.
 I use Webroot, and I also run Malwarebytes, Malwarebytes Anti-exploit, and Norton Power Eraser regularly. 

Free options: thanks to Nick C for pointing out that your ISP (e.g. Verizon, Comcast), workplace or college probably has some organization-wide licenses that will get you commercial software for free or at a reduced price.

Malware removal: see below if you have malware on your computer.  [1/22/16 note: I haven't reviewed that section in a long time.]
My current (1/22/16) backup software recommendations.

Use an online backup service. 
You probably won't be able to backup your entire PC (most hard drives are too large), but all of your important files will be protected.  I use Mozy (from storage giant EMC).  Carbonite is another big name -- I had an installation problem, so I gave up.  The best value is Acronis Online Backup, but the software is full of bugs and they have the worst tech support on Earth.  I gave up after trying to make it work for six months.  I also tried Dell Data Safe -- another disaster. 

Note that online backups don't do a full image of your hard drives.  In the event of a catastrophic event, you would have to reinstall Windows (etc.) -- and then restore your documents (pictures, music, etc.) from the online backup. Many hours of work.  Thanks to Nick C for pointing out that it can take a while (two weeks or even longer) before the online backup has completed an initial backup (due to the size of your files and speed of your Internet connection).

Save a daily disk image of your hard drive on an external hard drive.  A disk image is what it sounds like -- it can be used to completely restore your hard drive (or a new replacement hard drive) to a point in time.  What a timesaver if you get a virus or have a hard drive failure or corruption!  A Windows XP update once corrupted my Windows installation -- I restored my Windows folder from the prior day's image and was back up and running in 15 minutes.  I use Acronis True Image Home -- their user interface is horrendous, but I've had nothing but problems w/ their only real competitor and market leader, Norton Ghost.  Thanks to Nick C for suggesting that Windows 7 Backup is worth a look b/c it's free and already installed (if you're using Win7).  Windows 7 Backup has a limited feature set, though.

Pro and cons, amplifications and more recommendations: see Backups, below.
The below is more info and justification for the above recommendations.  Not worth reading for most people, except perhaps for the Backups section and Malware removal section.  I have heard time and time again -- you don't want a lecture from me, you just want to know what I think you should do!  :-)  1/2216 note: I haven't reviewed the below in a long time.
Security software explained.
Antivirus: detects and removes malware that does malicious things.  Typically, viruses and worms.
 
Anti-spyware: detects and removes software that collects information used mostly for marketing purposes and customizing content.  Much of the time, any damage caused by spyware is due to inadvertent conflicts w/ other software on your PC.  Some spyware is downright anti-social, though, and controls Web browsing, tries to sell you things, resists removal attempts, etc.  Some spyware can be considered beneficial -- the Google and Yahoo! toolbars, for example.  
 
Firewall: stops hackers from detecting the presence of your computer on the Internet.  A good software firewall (unlike Microsoft's) also stops unapproved outbound connections to the Internet -- i.e. keeps programs (both programs you've installed and malware) running on your PC from "phoning home" w/o your permission.  In addition, if you're running on a network, your router (think of a router as the equivalent of the electrical fuse box in your basement) will also include a hardware firewall that will increase your security by making your PCs harder to find.
 
All three types must be active on your system if you want to be protected.  The line is so blurred these days -- most software packages do at least some of all three.  Keep that in mind as you read below.

Security suites vs. individual software packages.
You should probably pick a highly-rated suite like Webroot Internet Security Essentials, Norton Internet Security 2010 or Norton 360.  Me?  I like getting best-of-breed pieces and cobbling them together.  It's not for the faint of heart or anyone who is a computer novice. 

Suite pros: components designed to work well together (hopefully, they actually do).
Individual package cons:  the packages may have overlapping functions -- so, they may need tuning. In addition, the packages have to be updated (and purchased, if not free) separately.

I used to use Norton, but dumped it b/c Norton 2007 was as bad as any virus -- they had really lost their way.  From what I have been reading, Norton 2009 (and later) puts them back up near the top of the heap.
Antivirus software. I just use Webroot.  I hate McAfee and Trend Micro products -- I have always had problems with them.
For those still using Norton and McAfee: a year's worth of updates to the virus detection tables comes w/ a new purchase of Norton and McAfee.  So, renewing your subscription after it expires is insane -- just buy a brand new version for the subscription price or less, and get a year's worth of updates for free.  Best Buy, Staples or buy.com will offer a great rebate offer on the Norton products w/in three weeks of today.  Note: my prior experience w/ Norton and McAfee antivirus software has been horrible -- but I have no recent experience.

Scumware (aka malware and spyware).  The best overall package that I've seen for keeping your PC free of pests is Webroot.

My #1 choice for removing malware from an already-infected PC is the free version of Malwarebytes (it sure does) to remove infestations on several PCs.  Please buy the Pro version and support them if it helps you.   Download to a USB flash key, boot into Safe Mode with Networking, install, update definitions and run.

My #1a choice for removing malware from an already-infected PC is Norton's free Power Eraser.  [It's dangerous -- techies only when it's time to click "Clean."]  Requires an Internet connection.  I just used it to clear a rootkit from a friend's PC that fooled Malwarebytes.

This is interesting, though I haven't had a chance to use it yet: Microsoft's Standalone System Sweeper.

Ditto: Kaspersky Rescue Disk 10.

I came across some promising-sounding malware removal tips courtesy of one of Fred Langa's readers, too -- but I haven't had a chance to try 'em out.

Backups.  Sooner or later, this will happen to you: fire, theft, flood, hard drive failure or corruption, or a PEBKAC error (Problem Exists Between Keyboard And Chair -- in other words, you'll screw up something and hose an important file).  How often should you back up your computer?  It's easy: you should back it up whenever you've changed the files enough that losing your changes would be a major pain or catastrophic. 

My basic recommendations are above.  Here is some more info, and some advice.

Online backup, recommended above:
Pros: schedule and forget about it -- but don't forget that your PC has to be on at the scheduled time!
Cons: can't backup a full image of your hard drive due to size limitations -- if your hard drives crashes or your PC is stolen, you'll have to start over from scratch and then restore your important files from the online backup.  Also, most backup systems save only a limited number of file versions or for a certain period of time -- if you accidentally delete a critical section of a spreadsheet or a folder of baby pictures and don't notice for a year, you're out of luck.  And, remember that you are trusting that the backup service is using a system that provides end-to-end encryption services.  If you're really paranoid, encrypt your files locally into an archive (e.g. .ZIP) and then let the online backup software upload the encrypted file(s).  [Under that scenario, I'd consider using a service like Dropbox -- a service I do not recommend using w/o encryption if your files are super-confidential.  Their flawed security methods recently made headlines -- handle encryption yourself.]

Disk imaging, recommended above:
Pros: the ability to restore your entire hard drive (or any file or folder) to the state it was in at the time of the imaging.  Also, you can save any image as long as you want -- just copy it to another hard drive or burn it to DVD -- that would allow you to recover, for example, a file you damaged five years ago w/o realizing it.
Cons: one-time outlay for a 2TB external drive of $90-ish (as I write this in August of 2011), and the cost of the imaging software (about $50) if you don't use a free alternative (like Windows 7 Backup).

Most of you won't consider this, but you should: keep some of your images offsite.  Suppose you use the online backup and imaging strategies I outlined above.  If your computer is stolen or your house burns down (God forbid), your external hard drive will probably suffer the same fate.   Since the online backup only has your important files, there may be some holes.  So, why not store, say, monthly images on another extra external hard drive (or on DVDs) and take it offsite?  Offsite could be work, a relative's house, or even your car in a pinch (beware of weather extremes). 

Pros: all of the pros for online and imaging, plus:  full protection from physical disasters and theft, as well as the ability to go back to a distant point in time.
Cons: You gotta give it some thought, keep track of old images and develop a rotation scheme, and buy another external hard drive(s) or extra DVDs. 

For the truly paranoid or people protecting business data: you could buy multiple external drives, network attached storage (NAS) or perform FTP snapshots to a server.  If you need to do this kind of thing and aren't a techie, you're better off finding an IT specialist to help you.