So, you just bought your child's safety seat and you aren't certain you installed it correctly. What do you do now?
Well, the first thing to do (assuming you have the correct seat for your child), is to read the manuals for both your vehicle and your child's safety seat. These provide vital information (such as instructions on how to lock your seatbelts and warnings about dangerous practices) as well as helpful tips.
Okay, you've read both manuals. Now what?
1. Select the correct harness height and strap tightness for your child.
If your child is in a booster, make certain the seatbelt guide is in the proper position. Rear facing children should have the harness at or below their shoulders; forward facing should be at or above. Booster seats should position the seatbelt comfortably over the shoulder. The manual for your seat will give particulars of where it needs to be as well as how to move the harness.
How tight should the harness be? It should be tight enough that you cannot pinch a fold in the harness webbing at the collarbone.
If you find your child's clothing requires loosing the harness to fit, that clothing should NOT be worn in the car seat. This is especially important during the winter when many parents tend to want to buckle their children in their seats with their heavy coats on (or use aftermarket products like the bunting bags). For warmth, use blankets on top of your child. If it's thicker than a blanket sleeper, it's too thick. Bunting bags tend to interfere with the routing of the harness.
2. Select the correct "belt path" for your child's seat.
(Convertible seats only!) The belt path is where the seatbelt or LATCH straps are threaded through the seat. Again, the manual for your seat will tell you where that is and which path you need to use.
3. Decide where and how you want to install your child's seat.
Any back-seat position is fine; traditionally the center is considered safest becase it's the furthest from the potential point of impact in a crash. However, just being in the back seat provides a great deal of protection, and there are times when it simply is not possible to have your child in the center.
The best position is the one in which you get the most secure installation. Sometimes that will be the center, sometimes it will be the side (also called "outboard".)
If your vehicle was manufactured after 2003 you will have LATCH - Lower Anchors and Tethers for Children. Many vehicles (and some child seats) do not allow you to use LATCH in the center. Check your owner's manuals to verify this as well as the weight limit for your lower anchors. (40-48 pounds is standard, a few companies defer to the child retraint manufacturer, and a couple of companies have higher limits.)
A LATCH-installed seat is not safer than seatbelt-installed seat. Use whichever system with which you get the best installation.
Remember that your child's seat can not be installed on a rear facing vehicle seat! There are a couple of vehicles on the road now that allow the vehicle seats to be used when rear facing. Use these ONLY with adults and older children not in car seats or boosters.
4. Install the seat using LATCH or the seat belt. (NEVER BOTH!)
If you are using the LATCH anchors rear facing, simply clip the anchor attachments to the lower anchors, then tighten. Look for the LATCH symbol, this is usually located near the lower anchors to help you locate them. You can also find this information in your vehicle's manual.
Universal LATCH symbol 
Universal Top Tether symbol
If forward facing, do the same, then attach the top tether to the approved tether anchor. Look for the tether anchor symbol.
Do not tether to anything other than an approved tether anchor.
To use the vehicle seat belt, thread the seatbelt through the appropriate belt path and buckle. Then tighten the belt as much as you can.
If you have locking latchplates (see FAQ), you'll want to make certain at this point that the buckle is actually holding the belt webbing parallel to itself. If it isn't, see if you can tighten the belt webbing more, pulling the buckle into the correct position.
For switchable retractors (see FAQ), once you get a good tight installation, go ahead an lock the belt. You will further tighten the seat after the belt is locked, but this will help make it a little easier.
When you are tightening the belt webbing, be it the seatbelt or the LATCH straps, pull the webbing parallel to itself. It is not necessary to be particularly strong or big to get a secure installation! By pulling the webbing parallel to itself, you are using the geometry of the design to do the hard work.
If forward facing, use your top tether if you have one. If you aren't certain you have one, you may be able to have it retrofitted. The most efficient way to find out if you can have top tether anchors retrofitted is to post a "tether anchor lookup" post at http://www.car-seat.org. You will need to know the make, model and year of your vehicle.
Most rear facing seats CAN NOT be tethered. As of 7/2008, only seats manufactured by Britax and Sunshine Kids can be tethered rear facing. However, if your seat is manufactured by Sunshine Kids or Britax and you are using it rear facing, by all means take advatage of that ability as long as it is not prohibited by your car's manual. (Some vehicles with advanced air bags do not allow bracing or rear facing tethering.)
Consult your manual for instructions on how to do this.
At this point your seat should be securely installed. A secure installation is one in which the child is properly secured in the seat, and the seat (using the correct method and belt path for your situation) is installed with less than one inch of movement in any direction.
If you are having trouble at this point for whatever reason, please use either the "contact us" form or the forums at http://www.car-seat.org to ask for further assistance.