A fuel cell is an electrochemical energy conversion device.  Most fuel cells convert the chemicals hydrogen and oxygen into water and through this process, produces electricity.

An example of an electrochemical device is a battery.  A battery has all of its chemicals stored inside, and it converts those chemicals into electricity.  This means that a battery eventually "goes dead" and you either throw it away or recharge it, since the chemicals within it are ‘used up’.

With a fuel cell, chemicals constantly flow into the cell so it never goes dead -- as long as there is a constant flow of chemicals into the cell, electricity flows out of it. 

Different types of fuel cells:

Solid oxide fuel cell (SOFC)
These fuel cells are best suited for large-scale stationary power generators that could provide electricity for factories or towns.  This type of fuel cell operates at very high temperatures (between 700 and 1,000 degrees Celsius).  This high temperature makes reliability a problem, because parts of the fuel cell can break down after cycling on and off repeatedly.  However, solid oxide fuel cells are very stable when in continuous use.  In fact, the SOFC has demonstrated the longest operating life of any fuel cell under certain operating conditions.  The high temperature also has an advantage: the steam produced by the fuel cell can be channeled into turbines to generate more electricity.  This process is called co-generation of heat and power (CHP) and it improves the overall efficiency of the system.

Alkaline fuel cell (AFC)
This is one of the oldest designs for fuel cells; the United States space program has used them since the 1960s.  The AFC is very susceptible to contamination, so it requires pure hydrogen and oxygen.  It is also very expensive, so this type of fuel cell is unlikely to be commercialized.

Molten-carbonate fuel cell (MCFC)
Like the SOFC, these fuel cells are also best suited for large stationary power generators. They operate at 600 degrees Celsius, so they can generate steam that can be used to generate more power.  They have a lower operating temperature than solid oxide fuel cells, which means they don't need such exotic materials.  This makes the design a little less expensive.

Phosphoric-acid fuel cell (PAFC)
The phosphoric-acid fuel cell has potential for use in small stationary power-generation systems.  It operates at a higher temperature than polymer exchange membrane fuel cells, so it has a longer warm-up time.  This makes it unsuitable for use in cars.

Direct-methanol fuel cell (DMFC)
Methanol fuel cells are comparable to a PEMFC in regards to operating temperature, but are not as efficient.  Also, the DMFC requires a relatively large amount of platinum to act as a catalyst, which makes these fuel cells expensive.

Automobile Fuel Cells

Polymer exchange membrane fuel cell or Proton exchange membrane fuel cell (PEMFC)
Most commonly used for transportation applications.  The PEMFC has a high power density and a relatively low operating temperature (ranging from 60 to 80 degrees Celsius, or 140 to 176 degrees Fahrenheit).  The low operating temperature means that it doesn't take very long for the fuel cell to warm up and begin generating electricity.

Sources:  howstuffworks.com, pbs.org and fueleconomy.gov.