Bikes can be transported on a roof rack, hitch rack, or trunk rack. Roof rack is expensive and difficult to install. Trunk rack is cheap, easy to install, and hard to use.
You can transport your bike three ways: on top of the car, behind the car on the trailer hitch (with the sub-variant of over the spare tire hitch), and tied to the trunk door by the strap-rack. I shopped for a fitting solution for my family -- two parents and three children of 3, 5 and 7 -- and didn't buy anything. Instead, we put the bikes inside the minivan. Here's why.
Roof-top racks have to be installed on the roof-top rails. Our Honda Odyssey LX doesn't have any, but the Odyssey EX does. Thule and Yakima, two major sports rack brands, carry roof rack bases ($200-$400 without installation). A rack for one bike costs $100-$150. I didn't figure out how to put the bike on a roof I cannot reach on my toes. People who bike and travel with bikes a lot say it's the best way in terms of bike safety and stability. Bike doesn't swing on the rack and there's less risk somebody will damage your bikes and your car. On the other hand, you have to watch for clearance, and it's not fuel-efficient.
Trailer-hitch rack requires a trailer hitch, which costs $200-300. The rack itself costs from $120 for 2-bike tilting rack to $350 (and up) for 4-bike swinging-with-the-bikes-on-rack. 5-bike rack is an existing product, but far less common than 4-bike. Speaking of brands, extensive and contradictive comparison tables are available. My conclusion: if you need a good rack (that is, travel a lot and care for your bike), get Thule or Yakima. If you don't, buy the one your local bike shop or Target carries, but make sure you can return it if it swings and rattles as you drive. All in all, hitch-mounted rack would cost $400-$600, and we decided we don't yet ride our bikes this much.
Trunk racks are cheaper. They start at $50 for a 2-bike rack. A trunk rack is held on place with several pair of straps, and one pair goes over the top of the trunk door and threads through the gap between the door and the car. In Honda Odyssey, there is a plastic wing glued to the top rim of the trunk door. You cannot use trunk rack on an Odyssey without roof rails. With roof rails, you can (allegedly inconveniently) strap the bike rack to the rails. I have not seen this contraption.
Instead, we folded down the back seat. Two grown-up bikes we put in the middle side-by-side with the front wheels between the seats in the middle row. Two kid's bikes and a kid's bike seat fit in the trunk easily. If we have to go more than a couple of miles, we need to take another car.
See also: trailer hitch.