There's one reason and one reason only to buy a used computer online, lack of money which is a good reason. For the purpose of this article, the term computer shall include laptops. This article will tell you what to look for and what not to look for in a used computer, step by step.
I would only consider buying a used computer if you can't afford to buy the computer you want new and can't afford a new computer at a level below that computer provided its features still have what you want. IE, if you can't afford to spend $750 for a new Core 2 Duo computer but can afford to spend $500 on a new Core Duo computer or $500 on a used Core 2 Duo, I would probably get the new Core Duo Computer unless the features on the used Core 2 Duo you found you couldn't do without. The main advantage of buying a computer used is the warranty and the ability to extend it, free technical help from the computer company and the fact that you know there is no spyware, adware and viruses on it and in all likelihood no pre-existing problems on the computer waiting to be discovered.
Once you decide you can only afford to get the computer you want used then the first thing I would do is look to to see if it's available online from a large company that sells refurbished computers with at least a 45-90 day warranty. Refurbished computers include computers whose boxes were opened and never used as well as returned computers with no problems and returned computers with problems. These companies usually buy refurbished computers in mass and test, repair and restore them as applicable to their original factory setting before they are resold, so you're not likely to get a stiff. If you are, you will be covered by their warranty. You usually can extend your warranty for a few years beyond the original 45-90 days, as long as it's before your original warranty expires.
The next step below that is buying your computer used online. Usually when you are buying a computer used online, there is no warranty. If you are not sure if a computer includes a certain feature on it, you need to contact the seller and have him confirm by email if it does or doesn't have that feature before you purchase it. Sellers are notorious for being deliberately vague on used computers. For instance, a few years ago I purchased a computer which the seller confirmed to me by email that it included an operating system and when I got it, of course it didn't but because I had this in writing he replaced it with one that had an operating system. If I didn't get him to confirm this, chances are he would have stated that nowhere in his sales description did he say an operating system was included and therefore he won't replace it. If you don't have a basic general knowledge of computers and you have a friend that does, I would have the friend check out the description before you make your purchase. Whatever you do, if you see the words "Sold as Is"- do not buy as that is the sellers way of saying you are stuck with it, once you receive it.
I would not purchase any computer that is not at least described as being in very good condition as most sellers have a tendency to describe the condition one level better then what it is. I also would not buy any computer that mentions any kind of problem in its sales listing even its been fixed as you're asking for trouble. You should make sure the memory, hard drive, operating system, size and CPU is listed and what you're looking for. Assuming everything is in order, I would insist that the seller cleans the computer of all spyware, adware and viruses, if applicable before they send it out. I once got a used laptop that I had to spend 5 hours cleaning that junk off it. Once you get the computer, run any security programs on it and if you see that the seller hasn't sent you a clean computer, either tell them to refund you for the the whole computer or tell him you'll keep it but you want a rebate for sending you an unclean computer.
One last thing before you make your purchase is to see what software is already on the computer and if there's something you don't want, ask the seller to uninstall it and if there's something you want kept on, tell the seller to leave it on. I would have the seller send you the computer UPS ground as its likelier to be damaged if sent by air, insured for the full purchase amount and requiring you to sign for it when delivered.
Once you get your computer I would as I said earlier, run any security programs you have to make sure there's no junk on it. I then would test the computer for its basic functions and once you have an internet connection established, download Belarc Advisor online which is a free download that reads what's exactly on your computer in detail to make sure everything you were told was on the computer is on the computer in terms of software, system, RAM, hardware, etc. I also would make sure you have an anti-virus program operational before you start surfing the web.