Blockages in chimneys occur for a variety of reasons. Take, for example, a normal fireplace flue. If this flue was not capped by a chimney cap, it was be extremely easy for birds, squirrels, or other critters to climb into the chimney bringing large amounts of nesting materials with them. This nesting material can completely block your flue, preventing the chimney from drafting properly. This material is extremely flammable; flues are not meant to contain fires only the products of combustion. If this nesting material were to catch on fire and the flue were to contain this fire, flue tiles could crack and the entire chimney could catch on fire.
Always, always, always unblock from the bottom. Never unblock a chimney from the top. In cases of nesting material there are corkscrew attachments that can be put onto cleaning rods to actually hook into the nesting material and lift it out.
If you are dealing with a chimney that is venting two different flues, say the furnace and the hot water heater, the white wall that is between the two flues may buckle and block the flue system. This can cause these appliances to vent improperly. You may have a carbon monoxide detector go off if this is the case because often times the gases that are the products of combustion from gas appliances cannot escape the chimney. In this case you would unclog the flue from the bottom. You may need to widen the bottom opening where the appliance connects in order to successful unblock the flue system in this case. Signs that a white wall has collapsed would be the appliance immediately shutting off or the carbon monoxide detectors going off.
The flue is not properly unblocked until one of two things occurs. First, if the flue has no bends or offset flue tiles, the flue is properly unblocked when you can tilt a mirror up the flue and see daylight (when the cap is removed, of course). If there is a bend or offset flue tiles, the flue is properly unclogged when you can run a sweeping rod the entire way up the flue system to the very top.
Clearing out a blocked chimney can be very expensive, especially if the sweep charges per hour and takes a long time to remove compact debris. The best way to avoid a blocked chimney is to install a chimney cap. This cap will prevent animals of all shapes and sizes from bringing any nesting material into your flue. This will prevent costly chimney repairs if a fire catches up in the flue system.