A vented dryer pulls in the surrounding air and heats it. The hot air is pushed through the drying clothes, then expelled through a vent that leads outside of the house. A condenser dryer doesn't need a vent because of differences in internal design. Instead of expelling the hot, moist air, the condenser dryer contains a heat exchanger, which removes the moisture from the hot air and "recycles" it, passing it back through the drying clothes. The excess water is either drained away or caught in a container that is later emptied.
The main advantage of condenser dryers is that you don't need a vent to use it--unlike a vented dryer, you can place it anywhere you like. This can make a condenser dryer valuable for people with a small living space, such as apartment dwellers. A condenser dryer can also be a good bet in situations that require very long vent runs, which can be inefficient and difficult to clean.
Condenser dryers have two types of cooling systems. Some condenser dryers have air cooling systems. The condenser dryer draws in air from the outside then expels the now heated air into the room (the air should be moisture-free, though, if the condenser dryer is working properly). Condenser dryers can heat a room significantly and may be best used in cool or moderate climates. You can purchase a water-cooled condenser dryer that won't have this problem, although you should consider your area's water supply before using an appliance that will draw extra water.
Condenser dryers need slightly more maintenance than vented dryers--the condensing unit must be cleaned about once a month to remove any lint. A poorly designed or malfunctioning condenser dryer can also let damp air into a room, which puts you at risk of growing mold on walls or carpets. Condenser dryers tend to be slightly less energy efficient overall than vented dryers.
Condenser dryers are part of combined washer/dryer machines--the same drum washes the clothes, then switches over to dry them. These machines are great if you're looking to save space in a small apartment or if you don't have a vent in your house or apartment. You can also "set and forget" your clothes, without the chore of switching clothes from the washer to the dryer. However, combination machines tend not to be able to dry as many clothes as they can wash, so be prepared to do several small loads of laundry if you want your clothes to be fully dry when you take them out.