What are different types of weights of an aircraft?
The different terms represent different weights or masses1. For each flight, the weights are taken into account for several reasons.

A brief description about these is below:

Manufacturer's empty weight (MEW)
Also called Manufacturer's Weight Empty (MWE) or Licensed Empty Weight

It is the weight of the aircraft "as built" and includes the weight of the structure, power plant, furnishings, installations, systems and other equipment that are considered an integral part of an aircraft.

This excludes any baggage, passengers, or usable fuel2.

Zero-fuel weight (ZFW)

This is the total weight of the airplane and all its contents (including unusable fuel), but excluding the total weight of the usable fuel on board.

As a flight progresses and fuel is consumed, the total weight of the airplane reduces, but the ZFW remains constant.

Maximum zero fuel weight (MZFW) is the maximum weight allowed before usable fuel and other specified usable agents (engine injection fluid, and other consumable propulsion agents) are loaded.

Operating empty weight (OEW) (Roughly equivalent to basic empty weight on light aircraft)

It is the basic weight of an aircraft including the crew, all fluids necessary for operation such as engine oil, engine coolant, water, unusable fuel and all operator items and equipment required for flight but excluding usable fuel and the payload.

Payload

It is the carrying capacity of an aircraft. It includes cargo, people, extra fuel. In the case of a commercial airliner, it may refer only to revenue-generating cargo or paying passengers.

Maximum takeoff weight (MTOW)

This is the maximum weight at which the pilot of the aircraft is allowed to attempt to take off3.

Regulated takeoff weight (RTOW)

Depending on different factors (e.g. flap setting, altitude, air temperature, length of runway), RTOW or maximum permissible takeoff weight varies for each takeoff. It can never be higher than MTOW. More information is on this answer.

Maximum landing weight (MLW)

This maximum weight at which an aircraft is permitted to land3.

The following image depicts takeoff weight components.


Image Source

Maximum ramp weight (MRW)
also called maximum taxi weight (MTW)

It is the maximum weight authorized for maneuvering (taxiing or towing) an aircraft on the ground.

Aircraft gross weight

It is the total aircraft weight at any moment during the flight or ground operation. This decreases during flight due to fuel and oil consumption.



1: As mentioned (below in comments by SentryRaven), several recent references use the term mass instead of weight. Having a Physics background, I tend to agree and know the difference between weight and mass. I did not use mass as most of the references I included referred mass as weight. However, using correct terminology, all the weights mean mass. Thanks SentryRaven for pointing it out.

2: What is the different between usable fuel and unusable fuel?
Usable fuel is the fuel on board an aircraft that can actually be used by its engines. The small amount of fuel that cannot be drained from the tanks is the unusable fuel. For calculation of range, usable fuel is used. For weight and balance total fuel (usable + unusable) is used.

3: This restriction is due to structural, design or operational limitations.