To add to the links above, a tropical wave is an area of showers and thunderstorms, usually associated with a large-scale area of low pressure. These are extremely common, especially this time of year in the northern hemisphere. For the most part, tropical wave and tropical disturbance are interchangeable, however there is a stage of development after the tropical wave and before a tropical depression (which has its definition in the link), and it is characterized by having 12 hours of persistent convection (thunderstorms) and a definitive low level circulation center. When these two criteria are met, the storm is officially a disturbance. A depression is a disturbance that has at least one closed isobar, and usually has maximum sustained winds of a least 20 mph.
As for other terms…I think the link covered it well for the most part, though their “track” graphic is a bit inaccurate, especially in the Indian Ocean.