A wrecker provides used parts for cars and trucks. Some wreckers will get the parts you want off one of their vehicles in stock while others will let you hunt for the parts you are after yourself. Second hand parts are usually a cheaper alternative to buying original factory parts.
Most wreckers will purchase cars. Generally they will only offer salvage value for a vehicle so if the car is running you would be better to try and sell privately first.
Most wreckers are able to pick up a car with their tow truck. This can be a good way to offload a vehicle that is just taking up space at your house.
Yes, auto parts recyclers generally offer the same services as wreckers do, they can also be known as auto dismantlers.