When talking about airplanes, a "tailwheel" aircraft, or a "taildragger" refers to the position of the landing gear on an airplane. An aircraft with a tailwheel, also known as conventional gear, is an aircraft designed with its two main landing gear placed forward of the center of gravity of the airplane, or in the front of the aircraft, with a single "tailwheel" in the rear of the airplane to support the tail of the plane.
The term conventional can be confusing, though, as the more commonly seen gear on light aircraft these days is tricycle gear and not tailwheel style. Nevertheless, the term conventional (which came from the fact that most airplanes long ago were designed with a tailwheel gear configuration) implies that the airplane has a tailwheel, as opposed to tricycle gear.
Aircraft with tricycle landing gear - where the two main landing gear are fixed behind the center of gravity, and the nose gear supports the nose of the aircraft in front - is more commonly seen in light aircraft today.
The Center of Gravity Challenge
Tailwheel aircraft are often thought of as more challenging - perhaps even more hazardous - to fly, than an airplane that has tricycle landing gear. Because the position of the center of gravity is located to the rear of the main gear, ground operations (mostly landing) can be more difficult in a tailwheel aircraft.
Less Forward Visibility
The tailwheel aircraft also sits with its nose higher than tricycle gear airplane, lowering forward visibility for the pilot during ground operations. It's more difficult to taxi without being able to see directly in front of you, which is why you'll often see pilots of tailwheel aircraft do S-turns while taxiing. And steering a tailwheel aircraft is different than steering a nosewheel aircraft since steering is accomplished from behind the pilot instead of in front.
But Wait... There are Benefits, Too
There are certainly benefits to a taildragger, as well. The nose-high attitude on the ground means that the propellors on tailwheel aircraft often have more clearance from the ground, making them better suited for grass or dirt runways. And they're often designed and configured for slow flight, making them easier to land on short runways. Many are high-design and are better suited for backcountry flying than nosewheel aircraft are. Tailwheel airplanes are without a doubt the favorite airplane among bush pilots.