Each light fixture will vary slightly in installation instructions, but the general idea is to remove the old fixture, connect the wires, and install the new light fixture. 

Step One - Pre-installation 
 
Ensure that the circuit breaker is tripped or in the off position. 

No power should be supplied to the existing light fixture.

Toggle the light switch to ensure that the existing light fixture does not power on.

Be sure to trip the circuit breaker. Sometimes, when only the switch is turn off, it appears to be safe to touch the wires but in fact the wires are "hot". 

Failing to trip the breaker could prove to be a very electrifying experience. 

Step Two - Remove the existing light fixture

When removing the old light fixture always take note of the wire connections of the existing fixture. 

White wire from fixture to white wire to ceiling. 
Black wire from fixture to black wire to ceiling, etc.... 

Save all caps, screws and bolts, just in case it is needed for the new light fixture.

Examine the existing mounting bracket to ensure that the new light can be installed with the existing bracket. Otherwise remove the bracket and install the bracket that came with the new fixture. 

If the wires from the new light fixture are not already stripped. Strip the plastic to allow about about 1/2 to 3/4 inch of exposed wire. 

Connect the bare wire to the ground wire or the grounding screw on the mounting bracket.

Green or Bare wires are often the ground. 
White and black are often the supply. 

It is important to segregate the wires and not to mix the colors of the wires. 

If you do mix the wires, the circuit breaker should trip and you will immediately know that you have wired the fixture incorrectly. 

Connect the white supply wire to the white supply wire from the ceiling or electrical box. 

Connect the black supply wire to the white supply wire from the ceiling or electrical box. 

From your notes review the previous wiring patterns. If there is a red wire from the ceiling it may be going to the switch, in this case connect this to the black supply wire of your fixture.

Twist the matching wires together and use the caps to secure the wires, the caps should cover all the exposed wire. If they do not, then you may need to reduce the amount of exposed wire.

Installing the New Electrical Fixture 

Depending on the type of light, the new light fixture will either be mounted with two mounting screws and a large center screw with a bolt.

Install the proper bulb for the light fixture.

Trip the circuit breaker back to the on position. 

Turn the light on to see if it works. 

If it works, complete the installation by installing the rest of the parts i.e. the light coverings. 

For Dome Lights that are flush to the ceiling

Following the directions above to install the mounting bracket and wire connections. 

Tie strings to both mounting screws. 

Thread the strings through the mounting holes. 

Pull on the strings and push on the fixture to guide the screws through the holes. 

When the screws are through the mounting holes, start screwing the screws to tighten fixture to the mounting bracket. 

Once the fixture is nice and snug you can cut the strings.
 
Handyman Help Info TIP:

When mounting the dome light fixture, you may want to use a piece of string to guide the mounting screw through the mounting holes of the fixture. 

Trouble Shooting - If your newly installed light fixture fails to work.

Double check the wiring, make sure the wire colors match.

Double check the wires from the fixture to the ceiling wires. Ensure that the wires from the fixture are twisted around the ceiling wires and are making contact.

Ensure that the caps are turned and twisted tightly around the connected wires.

Ensure that the circuit breaker is in the off position when you are handling the wires.

WARNING:

Be sure to switch off the circuit breaker before handling the wires.

This article was written for Handyman Help Info by Robert Huie who has obtained an Undergraduate Degree in Electrical Engineering and his Masters from George Washington University.