Lamps sold as one or the other - yet they could easily be both.
MR16 refers to a glass lamp enclosure with a Multifaceted Reflector (MR). The number indicates the diameter in 1/8 inch increments.
An MR16 lamp is therefore 2 inches in diameter, or approximately 50mm.
Lamps commonly sold as 'MR16' are usually those with two small pins on the back, the pins being GX5.3
GU10 is a type of bayonet connector which has two pins spaced 10mm apart. They are often used on MR16 lamps with aluminium reflectors which operate at mains voltage (230V or 240V).
A similar variant is GZ10 - the pins are identical, but the ceramic base has a square edge rather than a chamfer. These have dichroic reflectors.
A GZ10 lamp cannot be fitted into a GU10 socket, however a GU10 lamp will fit into a GZ10 socket. This is to prevent the overheating which would occur if a dichroic lamp was installed into a fixture which was designed to have an aluminium reflector lamp.
A connector which has two small pins spaced 5.3mm apart. Typically used on MR16 lamps which operate at 12 volts.
The connectors are (left-right) GX5.3, GU10 and GZ10
The pins on the GU10 and GZ10 are identical, however the ceramic base has a chamfered edge on the GU10 and a square edge on the GZ10.
The MR series of lamps has several of other sizes, such as MR11 (1.3/8 inch diamater) and MR8 (1 inch diameter).
The G series of connectors also has many variants, including G5 and G13 as used on fluorescent tubes, G4 as on small MR11 halogen lamps, and G24 which is used for fluorescent PL style lamps.