How to change your car's oil at home.

You will need:
Ratchet/Socket set (alternatively an open ended wrench set) !! Not a monkey or adjustable wrench.
A car jack and jack stands (alternatively wheel ramps)
A catch tank.
A fresh supply of oil.
A fresh oil filter.

If you have wheel ramps have someone help set the ramps in front of your tires so that they will sit properly.  If you're using a jack make sure you set the jack below part of the chassis, not a fuel line, exhaust pipe, wire hose, or some other relatively fragile piece of the car's underbody.  Never set the jack on any of the plastic body panels.

Raise the car so that it is at least 16-20 inches off the ground.  Place the jack stands under part of the chassis, raise them to full extent, then get out from under the vehicle and release the jack.  It should plop down on the stands.  Check the sturdiness by gently pushing on the fenders and checking for movement.  If you have wheel chocks, place them behind the rear wheels of the vehicle.

Locate the engine oil pan.  This should be directly below the engine which usually has some sort of letters or numbers on it, or numerous wires entering the top of it).  The engine oil pan will be somewhat large and look like the bottom of a big plastic tub.  Find a bolt usually located in the center of the oil pan.  It's probably going to be at an angle.  Be careful not to try changing the transmission oil, this process is considerably different, and you really don't wanna drain the transmission fluid.

Now pull the catch tank under the drain plug.  Find the socket of appropriate size for the plug.  Unscrew the bolt and set it aside as the oil spills out into the catch tank.  Allow the oil to completely finish draining.

Locate the oil filter, will be on the engine block usually in a hard to reach spot, because car manufacturers love their mechanics to feel pain.  Use your hands or other oil filter wrench to loosen the filter and drain it as well

Replace the oil drain plug.

Pour some fresh oil into the fresh oil filter and rub some oil into the seal/gasket of the filter.  Now screw in the new oil filter.  Only tighten to hand tightness, do not tighten with a wrench.

Now locate the oil fill cap on the top of the engine block.  Pour in the appropriate amount, close up the cap, check that everything is out of the engine bay and from underneath the car.  Use the jack to raise the car and remove the jackstands, then release the jack so that the car is back on the ground.  Check the oil level with the dipstick, see that it is full.  If not, add oil until it is.  Test drive the car around the block or so, check underneath the car to make sure no oil is leaking.

You're done.