Gas furnace kicks on then off frequently
WARNING:
Furnaces are expensive, complex pieces of equipment. If you don't have the proper tools and/or knowledge, it's often best to let the professionals handle maintenance and repair.
Gas Furnace Ignition Sequence:

Thermostat calls for heat.
Pilot gas valve opens.
Ignition control starts (spark or glow).
Pilot gas ignites.
Flame sensor detects pilot flame.
Main gas valve opens.
Main burners ignite.
Blower timer starts.
Flame sensor detects main burner flame.
Blower starts.

High Efficiency Gas Furnace Ignition Sequence:

Thermostat calls for heat.
Draft inducer starts.
Vacuum switch detects negative pressure.
Pilot valve opens.
Ignition control starts (spark or glow).
Pilot gas ignites.
Flame sensor detects pilot flame.
Main gas valve opens.
Main burners ignite.
Blower timer starts.
Flame sensor detects main burner flame.
Blower starts.

Ignition Problems:
Vacuum Switch Not Closing
In a high efficiency furnace, if the vacuum switch does not close after a certain timeout. The furnace will shut down, then possibly retry a few times depending on model.  If after a given number of tries the switch still does not close, the furnace will enter lock out. Once in lock out, the furnace has to be manually reset (depending on model).
The vacuum switch will look something like this...

And will have a rubber tube connecting it to the draft inducer. This part cannot be repaired, and must be replaced if it's faulty.
Pilot Not Igniting
If the pilot does not light, you'll first want to check to make sure there is something trying to ignite it.  Typically a spark or glow ignitor is used, so first you'll want to determine which is being used.
Glow Ignitor
A glow ignitor looks similar to this, though may not be visible without further disassembly.

It works by passing a current through it, causing it to heat up and glow. It should heat up enough to ignite the pilot gas. If it doesn't heat up, it should be replaced.
Spark Ignitor
A spark ignitor looks like this.

Take notice of the thick, often orange or red wire that is typically used to connect the ignitor to the ignition control module.  This device works by generating an electrical arc between the two probes, causing the pilot gas to ignite.  If there is no spark, you'll have to replace the ignition control module and/or the ignitor itself.  If the ignitor is not igniting the gas, but there is a spark. You can try cleaning the electrodes with fine steel wool, to remove any carbon buildup.
Pilot Not Proving
For a pilot to &quot;prove&quot;, it simply means that the flame sensor has sensed the pilot flame.  If there is no pilot or the sensor doesn't detect it, the furnace will often purge the system and then try again. Furnaces will often try a set number of times, before entering lock out.
If you can see the pilot flame, but the sensor is not detecting it. Try gently cleaning the sensor with fine steel wool, to remove any built up carbon. Also make sure the sensor is in the proper location with respect to the flame (Check the owners manual for proper placement), and adjust as necessary.  If that doesn't work, replace the sensor.
Main Burner Not Proving
If the main burner ignites, but the furnace shuts down before the blower starts.  You'll want to check the main flame (or rollout, high limit, etc) sensor(s).  If your furnace has error code indicators, check those and compare to the owners manual for translation.  In this case, you'll want to test and/or replace each sensor and/or contact an HVAC technician