Also known as innertubes, bike tubes are the rubber liners that fill and hold air inside your tire. Clincher tires have a space for the tube between the rim and the tire, and tubes are easy to swap in and out. Tubulars, or sew-ups, have tubes already encased within the tire. The two most important aspects of bike tubes are valve stem and tube materials.
Tubes are typically made out of butyl rubber, which is inexpensive (about $4-$11) and maintains air pressure well. Standard road tubes (700x23c) usually weigh about 95 grams, with lighter tubes like the Michelin Aircomp and Kenda Superlight weigh in around 75-85 grams. They use less material but they're also made a little more even than less expensive tubes. Unless you're a racer or serious biker, standard road tubes will do fine.
The other tube material is latex. It's the most expensive material, but is more elastic than butyl, which results in less rolling resistance (rolling resistance is measured in terms of heat loss and tire/tube deformation as the tire rotates) and a more efficient ride. They are also lighter, more resistant to puncture and pinch flats, but don't hold air pressure as well as butyl. Latex tubes will cost you about $20-$25.
There's a fair amount of elasticity in both tubes, so they usually come in a range of sizes like 700x18-23c. It's not terribly important to find the exact size tube for your tire. They'll still work well. Sometimes riders who want a thicker tube (thinking it will prevent some flats and be more puncture resistant) will use a tube that's a little bigger than their wheel size.
Although it's normal for all tubes to lose air over time (known as "air-porosity"), you may want to use a sealant as an extra precautionary measure. Tire sealants are a liquid compound you put inside a tire tube that seals tiny holes and punctures almost instantly. When you get a hole, the high pressure air inside the tube pushes the sealant through the hole at high speed. When the sealant meets lower pressure air (ie. outside air), it coagulates and forms a seal. It's pretty effective. Slime brand sealant is made for standard butyl tubes, but can't be inserted in tubes with Presta valves. You can, however, buy tubes with Slime pre-inserted or cut small piece out of your tube, apply the sealant that way and patch up the hole afterward.