We have to repeat this so often, I'm sort of hoping a moderator will make this a sticky that we can just point people to in the future.
How... when and why... to reboot your Canon DSLR.
If your camera is acting strange, giving mysterious problems, it might help to "reboot" it.
When to reboot your camera. Your camera powers up and seems normal some of the time, you can access the menu, but it sometimes seems to stick in one mode or another. Or it has some operational glitch that doesn't clear simply by turning the camera's power off and back on, or perhaps by removing the main rechargeable battery briefly. Or it won't format a memory card or doesn't write properly to the card (in this case, also use a flashlight to closely inspect the pins in the memory card socket and the socket end of the card itself, to be sure none are bent or damaged).
When it's unlikely to help. If you are seeing error codes, or image problems that might indicate a failing shutter or dirty sensor, rebooting is unlikely to help. If the camera simply doesn't power up it will be impossible to reboot it. Intermittent complete shutdowns also aren't usually correctible with a reboot (look for a loose battery grip or damaged/corroded battery or battery contacts... some early BG-E2 grips for 20D were simply were poorly fitted and lose contact, search for DIY fixes on the Internet).
Why reboot your camera. Modern DSLRs are highly computerized. They have memory, computer chips and process data, same as your desktop or laptop. Just as a computer sometimes needs to be rebooted to clear memory and remove stray strings of data, doing the same might solve a problem or small glitch in today's DSLR cameras.
Emphasis on might. There's simpy no guarantee this will fix any specific problem. But, what the heck! It's free and easy to try. If it works, great! If it doesn't, well you are just out a few minutes time and a little effort.
Disclaimer: This is a fairly easy thing to do and there's very little danger of anything going wrong. Still, if you undertake to reboot your own camera it is done at your own risk. So if you're at all unsure of your capabilities, you should have a trained professional repair technician look at the camera instead of trying these procedures.
How to reboot your camera, followed by some model-specific notes:
1. Turn off the camera at the power switch.
2. Remove all the batteries, both the rechargeable(s) and the little silver memory battery. The latter is often a CR2016 or CR2020 lithium or similar in a little tray alongside the main, rechargeable. Just sllide out the tray and remove the battery. Among other things, this "date/time" battery serves to maintain the clock and date in the camera while the camera is turned off (your computer has similar on the motherboard). It lasts a long time, up to five years typically, but does occasionally need to be replaced.
Once all the batteries are out, you have two options. One method is quicker. Let's start with that...
3. Turn the camera on. It won't actually power up because there are no batteries in it.
4. Press the shutter release button once. It won't fire, but this action should drain any remaining power in the circuitry immediately. Go to step 5 below.
Alternatively, if you prefer or the above doesn't work, the slower method: After steps 1 and 2...
3. Turn the camera on.
4. Let it sit for a long period of time, allowing any and all remaining power in the circuitry to gradually drain off. Some have had success letting the camera sit for just half an hour, 45 minutes or an hour. Others have had to let it sit overnight.
Whichever of the above methods you use, once that's done the next step is to check if you actually got the camera to reboot properly...
5. Turn the camera off.
6. Reinstall all the batteries. (6a thru 6d list some optional items to do and check.)
6a. It's a good idea to wipe the small, silver battery so there are no finger oils on it (see below, Cleaning Contacts). Try to handle it by the sides that don't make contact.
6b. If your camera is 4 or 5 years old, you might want to replace the small, silver memory battery with a new one at this time. They usually only cost a couple dollars (US) and are widely available. Just read the code off the side (or off the tray it fits into) and get an identically sized one. And be sure it's a "CR" lithium, not a nicad or alkaline.
6c. Might be a good time to clean the contacts for both the date/time and the rechargeable battery too (see below).
6d. While they are out, closely inspect the contacts and batteries for any wear and tear, or cracking/damage, or any sign of leaks. Replace any damaged or leaking battery. Do not use it. Any damaged contacts in the camera would call for a professional repair.
7. Once the batteries are reinstalled, turn the camera back on.
8. Go into the menu and check the date/time.
9. If you got a proper reboot, date and time should need to be reset.
10. If not, repeat the process above, starting with step 1. It might be necessary with your specific camera model to use an alternate method of steps 3 and 4.
11. Once you are sure you got a good reboot, retest the camera and see if the glitch you noted earlier has cleared. If so, great. You're done.
12. If not (or in the unlikely event that you cannot get the camera to reboot) you might instead try updating or replacing the camera's firmware, which can be downloaded from the Canon website. Be sure to very, very carefully follow the instructions there as this is a riskier procedure. Most critically, check that the download is complete (checksum), have the camera's batteries fully charged, and don't touch any buttons or turn the camera off during the firmware installation. Interrupting that installation or installing an incomplete FIR file can "brick" a camera and require it be sent in for repair. If neither the reboot nor a firmware update solve the problem, you should have the camera checked by a qualified repair technician.
Important! This reboot procedure also might lose some of your Menu and Custom Function settings, it might reset them back to the factory defaults. Be sure to check those before taking the camera out to shoot.
Cleaning Contacts: A few drops of isopropyl alcohol (inexpensive and widely available 70% "rubbing" alcohol is fine) on a clean cloth will remove any oils, which are the most likely thing to cause problems with batteries. An alternative is to use disposable computer keyboard/monitor wipes, most of which are pre-moistened with isopropyl alcohol (check the label). Some of the contacts in most Canon appear to be gold plated so shouldn't be "scrubbed" with anything abrassive that might damage the soft gold coating. Also, gold doesn't corrode (that's why they use it), so you shouldn't need a strong deoxidizer, either.
Note: Most Canon models can, in fact, be used without the date/time memory battery in place, or if it is dead, powered only by the rechargeable main battery (or a substitute, such as the AA alkaline pack). Any images made while the date/time memory battery is out or inoperable will have incorrect or no date and time recorded in the EXIF metadata.
Canon DSLR model-specific notes:
60D - This model cannot be rebooted in the manner described above. The memory battery is an integral part of the main board inside of the camera and is not user accessible. Extensive disassembly is required to access it and there are no instructions regarding removal/replacement of the date/time battery in the 60D user manual. Most people should leave this to a trained repair tech and not attempt it themselves. Presumably, this would apply to the 60Da, as well.
xxxD/Rebel/Kiss series - The first few of these models have user interchangeable memory battery, same as 10D throught 50D below, and can be rebooted as described above. Those are 300D/Digital Rebel, 350D/Rebel XT, and 400D/Rebel XTi models. Beginning with 450D/Rebel XS and later models, the small memory batteries are not user accessable, so those models cannot be rebooted as described above.
10D through 50D, 7D - The memory battery is in a small drawer alongside the main rechargeable in the bottom of the camera. First remove and set aside the main rechargeable or any accessory battery/vertical grip needs to be removed to access the compartment. Then use your fingernail to slide out the "tray". AFAIK, all these models use this arrangement. It's possible some earlier models (D60, D30) do too, though I don't know this for a fact.
5D, 5D Mark II and 5D Mark III - The small memory battery for these models is not located alongside the main/rechargeable battery. It's in a tray located on the lefthand side of the camera, right near the bottom of the rubber covers over the USB, PC, etc. sockets. Lift those covers out of the way, then look for a small screw. You need a precision Phillips screwdriver (probably size 0) to remove that retaining screw, before you can slide out the tray and access the battery. Be very careful not to lose the tiny screw!
6D and 70D AFAIK from checking in their manuals, these are same as 60D and more recent Rebel/xxxD series... memory batteries are not user accessible, so there is no means of rebooting the camera as described above. If I am wrong, please let me know.
1D series - First slide out and set aside the main rechargeable battery. Then turn camera upside down. The small memory battery is located just inside the top of the battery compartment and slides out. AFAIK, all models use similar arrangement, though some have a retaining screw that needs to be removed.
Helpful reference: In all that I have seen where it is possible to access and remove the memory battery, replacement of the small time/date battery is nicely illustrated in the specific camera's manual (a pdf of which can be downloaded free from the Canon website, if you don't have the printed copy).
Add'l Notes:

Pay very close attention to the orientation of the button battery when you pull it...
You'll make yourself crazy wondering what's wrong if you put it back in the wrong orientation!
Good point!

I couldn't see any mention of it but it's worth mentioning when removing and reinserting the memory battery, not to use metal tweezers. Plastic ones are ok of course.
Another good point! (I'm not going to ask how you know that )
I'll add more notes or correct any errors, if anyone cares to contribute suggestions.