A quick list:
- Maximum speed with flaps at maximum deflection (v$_{FE}$)
- Maximum speed with flaps at take-off setting. Fly faster and the flaps might be torn off.
- Maximum speed with gear down (v$_{LE}$). Here not the gear itself, but the open gear doors will fail if that speed is exceeded.
- Maximum speed for full control deflections (maneuvering speed, v$_A$). Yank the stick to full deflection at a higher speed and risk overstressing the structure.
- Maximum speed in gusty weather (v$_B$). Here it is assumed that the aircraft happens to fly into a vertical gust of 50 (FAR part 23) or even 65 (FAR part 25) ft/s, which will greatly increase the lift force on the wings. Above that speed such a gust will overstress the wings.
- Maximum operating speed (v$_{MO}$)
- Design cruising speed (v$_C$)
- Maximum dive speed (v$_D$). This is the speed reached when at v$_C$ the aircraft enters a shallow dive and the pilot needs 20 seconds to react.
- Maximum operating Mach number (M$_{MO}$)
- Design cruise Mach number (M$_C$)
- Maximum dive Mach number (M$_D$). At least 0.05 Mach above v$_C$.
And that is only what is listed in the certification regulations. You can also distinguish between a maximum speed at which the landing gear is operated (v$_{LO}$) and the aircraft is flown with the extended gear, since large doors open during operation but are closed again when the gear is extended. And yes, for aircraft with retractable landing lights, there is even a dedicated maximum speed at which the landing lights are operated (v$_{LLO}$). As usual, Wikipedia even lists a few more.
Aircraft have several limits, depending on altitude. At low level, the dynamic pressure limits how fast they can go, mostly because of structural loads. At high speed, a deflected aileron will twist the wing, reducing its effectiveness. Better limit the speed to a point where the aileron is still effective.
Higher up, it is the Mach number which might limit your speed. If the aircraft is not designed for transsonic flight, Mach tuck and buffeting will set a clear limit, and the maximum Mach speeds are defined to prevent the airplane from venturing into that territory.
Heat loads might pose another limit. The F-16 is limited by the heat loads on its polycarbonate canopy and must not go faster than 810 KIAS at low altitude.
And, of course, there is always flutter. However, for certification it must be demonstrated that no flutter will occur within the allowed speed range +20%.