Rear- and front-facing car seats, booster seats: safety and usability.
All American states have laws regulating car seats for children. Alabama requires children up to 6 years old restrained and children up to 4 in the car seat or booster. New Jersey regulates both age and weight for car seat and booster (8 years or 80 pounds, whichever is later) and requires everybody restrained. All other states fall in between. Most states allow the exception for taxi, but it's still safer to have a baby or a toddler appropriately restrained.
Manufacturers typically don't recommend using the car seat for more than five years. Good car seats are expensive, but worth it. Hand-me-downs are OK if you can trust the previous owner it never was in a crush. By the way, most thrift shops will not accept used car seats.
Price range for different new seats:
Rear-facing infant seat: $55-$200
Convertible infant/toddler: $55-$250
Convertible toddler/booster:$60-$90
No back booster: $20-$40
High back booster: $70-$270
Various price and safety comparison data is available. Note that even most expensive car seat costs only a small fraction of the price of the car, and car crashes are one of a few real dangers of the modern life. Of course it doesn't mean the most expensive is the best.
Marketing terminology is sometimes confusing. Do you need a baby seat or a toddler seat? Convertible or flexible? Free-base or one-pice? There is no clear established nomenclature, so don't get intimidated and use your common sense.
Car seat is a baby-size chair that is kept in it's place in the car with the seat belt and sometimes additional belts and hooks that come with it (relatively new LATCH standard). They are sometimes tricky to install properly. Police stations, fire stations and some hospitals provide free car seat installation, and some hospitals discharge newborns only after car seat inspection. Car seats have their own set of straps, or harnesses -- usually three-point for the rear-facing only seats and five-point for forward-facing and convertible. In a booster, child uses regular safety belt. All seats and boosters are used with both lap and shoulder belt. Car seats for babies up to 1 year old or 20 pounds or 26-32" tall are rear-facing. Children up to 4-5 years or 45-65 pounds or 40-49" tall can ride in forward-facing seats. Older children should use booster seats until they are 90-100" and weigh 80-100 pounds.
The bigger and heavier the seat the better -- it has more material to absorb the impact. There are two types of rear-facing seats: ones that can convert to forward-facing position, larger, heavier and more expensive, and ones that don't: they are smaller and cheaper, and can be easily removed from the car and used as a portable cradle. I used both types. If you can fit the bigger one in the car I'd strongly suggest getting the large convertible -- Britax is the gold standard of safety. The basket-cradle is inconvenient anyway.
Convertible seats usually have 5-point harness system: two points at shoulders, two on hips and one in the crotch. Several slots for the shoulder straps let you adjust the straps as the child grows. Always use the slots at shoulder level or the closest higher. Some seats have an additional harnessing/protecting shield, either a padded over-the-head tray-like bar or a T-shaped piece that go over the lap.
Booster seats come seat-only or seat-and-back. The former is just a seat, the latter has high back with wings that protect the head and a slot to thread the seatbelt through at shoulder level. Some seat-and-back models (Britax, Graco, Cosco) allow growth readjustment. There are also forward facing car seat/booster convertibles which are cheap but not especially good at either function. They are lightweight and handy if you need the car seat only occasionally.
All in all, if you drive a lot, get a heavy convertible car seat for the first 3-4 years and an adjustable high back booster for the next 3-5. If you don't own the car, you still need a car seat once in a while. Get a rear-facing seat for the first year and a seat/booster convertible for the next three.
See also: children, baby carrier, newborn baby, jogging stroller.