A conventional-gear airplane — a 'taildragger' or 'tailwheel' airplane — carries its third wheel at the tail, behind the two main wheels. A tricycle-gear airplane carries its third wheel at the nose, ahead of the mains. The term 'conventional' is historical: nearly every airplane built before the late 1940s used tailwheel gear, so it was the convention. Tricycle gear only became dominant after WWII.
The single most important difference is the location of the center of gravity (CG) relative to the main wheels:
This one geometric fact is the source of nearly everything that makes a taildragger demanding: ground loops, the need for active rudder, the reduced forward visibility, and the discipline of flying the airplane all the way to a stop.
Flying a tailwheel airplane as PIC requires a one-time logbook endorsement under 14 CFR 61.31(i) (covered in detail later). There is no checkride and no knowledge test — your instructor simply trains you to proficiency and signs your logbook. But 'proficiency' is a high bar in a taildragger, and that is exactly why this course exists.