I'll put it VERY very simply for now, as judging by your questions, there is still a lot of research to be done. But this should get you started.
1) Part-66 - is the regulations under which, an engineer can certify aircraft work.
Part-145 - is the regulations under which an aircraft is maintained at a maintenance facility.
Part 147 - is the regulations under which theoretical material is taught/examined i.e Part-66 license modules, aircraft type courses.
2) CAT A - Engineer can certify 'limited & simple' tasks which the engineer has carried out personally. i.e wheels, brakes, light bulbs, cabin defects.
CAT B1 - Engineer can certify all engine, airframe and electrical work carried out by anyone. (including a lot of avionic tasks too!)
CAT B2 - Engineer can certify all electrical & avionic system work carried out by anyone.
CAT C - Engineer can release an aircraft from a major maintenance check. CAT C is granted in time and requires no extra study.
CAT A & CAT B1 are further divided into :-
A1 & B1.1 - Turbine engined aircraft
A2 & B1.2 - Piston engined aircraft
A3 & B1.3 - Turbine engined helicopters
A4 & B1.4 - Piston engined helicopters
B2 is applicable across all aircraft & helicopters (probably because you don't need to know as much
3) A person is eligible to hold a CAT A, B1 or B2 license after completing all of the applicable multiple choice & essay style examinations, and proving, in the style of a signed logbook, the amount of time required to hold the license type. (the amout of time varies depending on the license being applied for, and how you go about sitting the exams)
To actually sign off a maintenance task you will have to complete a type course for the applicable aircraft, then obtain a company approval.
Hope all this helps, no doubt someone will post pages and pages of regulations to demonstrate their knowlage, but that will probably only confuse you just now!