Keep it
Sometimes having a spare computer around is a good idea. You can set it up for a single task like holding your family pictures or streaming media to your TV. You could also keep it around for visitors or kids to use when they visit.
Give it to a family member
For years I would hand my old computer down to a kid or family member. This is especially a good idea if that family member’s computing needs are less than your own. I have an old 2001 iMac G4 that is on its third life with a family member who uses its modem every day.
Sell it
If you computer is new enough it still may have value for selling. A quick search on ebay.com will tell you what other people think your computer is worth. Macs keep a high percentage of their value over time. It may be worth your trouble. I have sold old mac and shipped them as far away as Korea and Japan. Remember to consider the packing cost in your calculation. Older computers are heavy. There are other sites like macsale.com and gazelle.com that make it easy to sell your old computers.
Donate it
Check with your local church or school to see if they accept computer donations. School districts may have high requirements for their donations. Check also with your local Goodwill, Salvation Army, or mission to see if they will accept your donation. This is a better option than recycling if the computer still functions.
Recycle it
Computers contain nasty chemicals that make it unsafe and in some cases illegal to just throw in the landfill. Proper disposal is required. You can take your old computer to a reputable recycling center where the parts can be safely separated and disposed. When you buy a new computer from Apple or Dell they offer programs where you can recycle your old computer for free. They send you a FedEx tag so they pay the return shipping. This works well if you have the packing materials from your new purchase that you can reuse. I recently learned that Best Buy will accept old computers for recycling even if you did not buy the new computer from them. You just bring them in to their Geek Squad counter and drop them off.
Fix it
It really depends on the underlying reason for why you are getting a new computer. The expected functional lifespan of a desktop computer is increasing every year. I typically consider four years to be my expected time with a computer. I also use that as a factor in calculating the cost of a computer. That makes a $1000 computer cost $21/month when you think about it like that. At some point the cost of fixing the computer exceeds the cost of replacing it. If you are handy there may be 
something you can do. Most computers today have consolidated to the point where everything is on the motherboard. If the problem is not in software, RAM, or the hard drive, then its probably not worth fixing. I have been known to take parts out of one computer to extend the life of another. I recently removed the CD-ROM from one iMac that had a bad mother board and installed it into another iMac that had been fed pennies. Surprisingly, iMacs don’t care for pennies. The result is a 15 year old computer that is ready for a new owner
Before you part with your computer…
Before parting with your computer make sure that you have gotten all your stuff off the old hard drive onto the new computer. Both Mac OS and Windows offer migration tools to help you move from one computer to another computer. These only work if you a disciplined and keep all your stuff in your home folder. When in doubt I recommend backing or cloning the entire hard drive. If your new computer has an extra drive bay and is compatible you can put the old hard drive in the new computer and use it for extra space. You can also buy USB kits to convert your old hard drive into an external hard drive to use for backing up files.
Wait
Once you are sure you have everything… wait. There’s nothing like a cooling off period between when you get set up on your new computer and you part with the old one. This is smart as you get used to the new system to still have the old computer as a fallback. If you use your computer for cyclical business needs then you may want to wait one cycle. For example, wait until you get your monthly bills out.
Sanitize
I always recommend erasing the hard drive of your computer before parting with it. This is the only way to be sure that you are not passing on your personal data or, in the case of Windows, a virus. On a mac this is as easy as booting your computer from the USB stick or DVD that came with it and then using the Disk Utility to erase the hard drive. I never leave software installed on a computer unless I am sell that software with the computer. And even then, I prefer to reinstall it clean. This does require that you have the orignal installation disks. This is getting to be more problematic in a world with App Stores. Most Dell computers have their Windows license number as a sticker on the side somewhere. All you need is a copy of that version of Windows to reinstall. If you don’t have a copy of Windows, there’s always Ubuntu which is free for personal use and makes a good option for older computers. Mac computers came with a license to the OS when you bought it. Check MacTracker.ca to lookup that version if you are unsure. Depending on who is receiving your computer may also dictate what software can go on it. In the end it may be a good option to leave the hard drive blank and then the recipient worry about the OS. When erasing your hard drive be sure to choose the security erase option. This rewrites the hard drive several times to protect from forensic dynostatic recovery of the data.
Remove
If you are recycling the computer some places will not accept a computer with a hard drive. You can remove the hard drive before recycling to be safe. There are services that will shred your hard drive for ultimate protection but that’s probably overkill. If I want to destroy a hard drive it’s usually as easy as opening the case with a screwdriver and shattering the platters.
RAM
The hard drive is not the only thing to be salvaged from your computer. Depending on how old your computer is, it may have RAM that is of value to you or someone else. RAM can be sold on ebay. RAM does not retain any of your personal information once the computer is powered off.
Arts and Crafts
Another option for some people is to turn their former business tool into a piece of art or furniture. Old Macintosh Plus have been turned into fish tanks while iMac G3 can be turned into a cat bed.