DCS World: This is a free package that is the foundation that DCS runs on. It includes all of the air, sea and ground models, AI, etc. etc. This is free, and includes an Su-25T that you can fly. All current DCS modules need this program to work.
DCS Black Shark: This was the original DCS, and allows you to fly a realistic recreation of the Ka-50 Black Shark. This is NOT compatible with DCS: World, you need to purchase...
DCS Black Shark 2: This is the same thing as Black Shark, but updated to be compatible with DCS: World. You can buy it at a discount if you own DCS: Black Shark 1.
DCS A-10C: This module allows you to fly a realistic recreation of a Suite 3 A-10C.
DCS P-51D: This module allows you to fly a realistic recreation of a P-51D Mustang
Flaming Cliffs 3: This is an update of Flaming Cliffs 1 (LOMAC V1.11/12) and Flaming Cliffs 2, adding the ability to fly the F-15C, A-10A, Su-27, Su-33, MiG-29A/C and Su-25 in DCS World. These aircraft are not as accurate as their "DCS" series counterparts. They have simplified avionics and flight models. Due to a previous business arrangement with Ubisoft, you are required to have a copy of a previous version of Flaming Cliff installed before you can install Flaming Cliffs 3. Lock On: Modern Air Combat, LOMAC:Flaming Cliffs, or Flaming Cliffs 2 will all satisfy this requirement.
To answer your question, the A-10A in FC3 is not the same as the A-10C. The A-10C has an advanced flight model, and simulates various systems and avionics such as displays, navigation and target computers, and datalinks that the FC3 A-10A only approximates. All DCS aircraft also have fully interactive, clickable cockpits while FC3 aircraft do not.
There will be a series of separate modules that model each of the FC3 aircraft with advanced flight models, but keep the simplified avionics. These modules will not require an installation of a previous Flaming Cliffs version.
It is my understanding people who have purchased FC3 will also receive these upgrades, but I may be wrong. I also believe I heard that in the future, Eagle Dynamics may develop these airframes to full DCS standards (AFM, full avionics, clickable cockpits), but again, I cannot locate an official statement to verify this.
DCS Combined Arms: This module allows you to control ground forces in DCS World. Options available to you are the ability to control individual tanks and SAMs, strategically control groups from the map, or act as a forward air controller. As a FAC you can mark targets with smoke, IR laser, or targeting lasers, as well as control vehicles.
THIRD PARTY MODULES
DCS "Huey": This module allows you to fly a realistic recreation of a UH-1H Helicopter. Features planned for the final release of this aircraft include multiple player pilots per airframe simultaneously over the internet (I.E. A friend operating the door guns while you fly), as well as a sling load. This is also the first "3rd Party" add on to be released. It was not developed by Eagle Dynamics, but by Belsimtek. ED sells this module for them through its online store.
DCS MiG-21: This is a module in development by Studio Beczl that will allow you to fly a realistic recreation of a MiG-21bis "Fishbed". This module has yet to be released.
DCS Hawk: With this module you will be able to fly an accurate recreation of a BAE Hawk light attack/ trainer. This module is still under development by Virtual European Air Operations.
DCS T-2 Buckeye: This module will be realistic representation of a North American T-2 Buckeye light attack/trainer. This module is still under development by Razbam studios.
DCS F/A-18E: This will be a realistic F/A-18E Super Hornet. It is still in the early stages of development. It will be finished by Coretex Designs.
DCS F-35A: This will be a "realistic" (as much as possible) recreation of the Lockheed Martin F-35A Joint Strike Fighter. Development by Kinney Interactive is just now starting, or will start soon.
DCS L-39: An accurate recreation of the Aero L-39ZO/ZA and dearmed variants. This module is in development by the Virtual Patriots Jet Team
DCS F-104G: An accurate recreation of a Lockheed F-104G Starfighter, in development by Mil-sim Projects.