The safest place to install a child safety seat is in the center rear position.
Once installed, the child safety seat should not move more than an inch side to side. To check this, grab the seat at the point where the seatbelt goes through it and shake it. If the seatbelt loosens up, it's not tight enough.
To get the seat in tightly, try this trick (may take two people). Put the car seat in the vehicle, route the seat belt through the designated path (most car seats will have stickers to show you where the seat belt should go for rear-facing or forward facing seats) and buckle the seat belt. Put all of your weight into the seat by climbing in it, putting your knees in it, or pushing down on the seat with one hand. With your free hand, take up all of the slack from the lap portion of the seatbelt. Test the seat to see if it's in tight enough.
What's a locking clip and do I need to use one? A locking clip is a metal object shaped like an H or T, and comes with new safety seats. If you're installing your child's car seat behind the driver or passenger seats in the back, you may need to use one. Pull the shoulder portion of the seatbelt all the way out until no more comes out. Slowly feed the seatbelt back in. If you can hear little clicks, or feel it "ratchet", then you probably don't need to use one. If the seatbelt freely goes back in or catches only every once in a while, it's likely you need to use one. Read your manufacturer's instructions for specific details.
Never attempt to secure a child safety seat with more than one seatbelt. It may seem like a good idea to use both the middle and passenger seatbelts to tighten down the seat, but it is dangerous.
Do not put a towel between the safety seat base and the vehicle seat unless you are using a rolled one to achieve a 45-degree angle. Many parents try to protect their seats, or achieve a tighter fit by using a towel under the safety seat. This can be dangerous as well. A towel puts too much space between the car seat and vehicle seat, and may actually make it easier for the child seat to slide out of position. Instead, try using a sheet of rubber shelf liner. Most dollar stores or grocery stores carry them. The rubber shelf liner sheets are extremely thin, and provide a good gripping surface, especially on leather seats.