Here's a basic breakdown of the differences between different auto warranties:
The Basic Warranty -- otherwise known as the manufacturer's warranty. These usually come with new cars and cover most of the components in a car that might break down during the early years through normal wear-and-tear. These usually do not cover tire or battery replacement, and you can forget about it if you misuse the car, do anything that may void the warranty, or get into an accident.
Dealership Warranty -- this is a warranty not from the manufacturer but sold by the dealership. There may be specific stipulations on dealership warranties, such as where you have to go to get your car serviced or maintained, and generally if a component in your car does breakdown, you will have to go to a specific dealership for the warranty to be honored.
The Extended Warranty -- the extended warranty is also known as the aftermarket warranty. These warranties extend the coverage on a vehicle for an additional period of time after the manufacturer's warranty has expired. Not all car models are eligible for extended warranties; it depends entirely on the company. It is best to buy an extended warranty earlier on, before your first warranty has expired, because the longer you wait and the older your car becomes, the more expensive the extended warranty will be. Extended warranties may also cover fewer things than the original warranty did.
Basic Warranty - this covers pretty much the entire automobile except tires and batteries. It includes: suspension, electronics, steering and exhaust. Remember that warranties only kick in if any of the above malfunctions through normal wear and tear.
Powertrain Warranty -- by powertrain they mean the mechanical components of the car that produces and channels kinetic energy. It generally includes: transmission, drive shafts, axle and clutch; but if something like the engine breaks down, the warranty company will not foot the bill for a new engine.
Rust and Corrosion Coverage -- as the term implies, this covers you in the event your car gets a bad case of rust. More specifically, it is unlikely the warranty company will replace anything unless the rust eats a hole through the metal. This sort of thing can also be covered through insurance.
Emissions Warranty -- this is a completely separate warranty that covers any emissions problems you may experience up to a certain number of years/miles. This may be covered under the Basic Warranty arranty, but not always.
Warranties may include all of the above or some of the above. The important thing is to closely review the fine print in your warranty contract. And just because the terms seem great does not mean the company is trustworthy. Do your research before signing anything.