Sp3-hybrid
When a carbon atom is connected to four neighbors by single bonds, chemists call it sp3-hybridized. Molecules with only sp3-hybridized carbons are saturated compounds, and a saturated hydrocarbon that contains only hydrogen and carbon is called an alkane. Typically the angles between the four bonds surrounding an sp3-hybridized carbon is ~109.5 degrees. If the carbons are forced to adopt an arrangement with smaller bond angles, the compound becomes higher in energy and thus more unstable. This effect is called bond-angle strain.
Sp2-hybrid
Sp2-hybridized carbons have two single bonds and a double bond. Atoms at the two ends of a single bond can rotate while atoms at either end of a double bond cannot, so molecules are free to rotate around single bonds but not double bonds. The bond angles around an sp2-hybridized carbon are usually about ~120 degrees or so. Molecules that contain a double bond between two sp2-hybridized carbons are alkenes and are said to be unsaturated. Unsaturated fats, for example, contain carbon-carbon double bonds, whereas saturated fats do not.
Sp-hybrid
Sp-hybridized carbons have either a single bond and a triple bond or two double bonds. They exhibit a linear configuration, meaning the triple bond and single bond are ~180 degrees to each other. Compounds that contain two carbons triple-bonded to each other are called alkynes; acetylene, a fuel used in welding torches, is the simplest example. Another more interesting compound with an sp-hybridized compound is hydrogen cyanide, an infamous poison that's also useful in organic synthesis.
Degrees of Unsaturation
There's a handy trick you can use to figure out how many double or triple bonds and rings a compound contains. First, take the number of carbons, multiply this number by 2 and add 2. Next, subtract 1 for each halide (bromine, chlorine, etc.) present in the molecule, and add 1 for each nitrogen atom. Now subtract the number of hydrogens actually present in the molecule and divide the result by 2. The number you get is your degrees of unsaturation. Each ring or double bond is equal to one degree of unsaturation, while each triple bond is equal to two degrees of unsaturation. If the molecular formula is C5H12, for example, there are 0 degrees of unsaturation. C5H10, by contrast, must have either a ring or a double bond.