ArmA2 is a military simulator more than it's a game. You can fight as a soldier or part of a team or lead a team or drive a tank or just gun in one or fly helicopters or planes or anything really. :) ArmA is a super expansive game so be patient in learning it. As with any game also who you play with determines how good a game it will be, so be prepared to find unbelievably horrible public online play before you find a really solid group of guys to play with. :) It's totally worth it.
The free version is the full ArmA2 game without the single player campaign/scenarios and with lowered quality graphics (all the textures are blurry instead of crisp HD awesomeness they usually are). The game it otherwise complete and the editor is fully functional.
The content you'll be missing is the aforementioned campaign stuff. The Operation Arrowhead expansion, which brings in desert terrain and the US Army units. The British Armed Forces DLC (BAF) which brings in the British units and some new features like engineers that can repair vehicles. The Private Military Company DLC (PMC) which brings in video playback in game and Blackwater/Xe type units.
If you upgrade to Combined Operations (Which is the ArmA2 game + Operation Arrowhead expansion) you'll get "Lite" versions of BAF and PMC. So all the content (minus the missions) and blurry units. You can then buy the DLCs on their own for $10 each or so or get "Reinforcements" which is a DVD with BAF + PMC + OA "Lite".
Personally I'd suggest one of two upgrade paths:
1. Buy ArmA X
directly from BIS. It's $60, so seemingly expensive, but you get EVERY ArmA game. OFP/A1/A2/OA + all expansions and DLCs for. All downloadable with the games and keys stored online for you. No DRM.
2. Or else I'd suggest getting
Combined Operations then the two DLCs. That'll be $55 or so, $30 for CO + $20 for the DLCs. Really, ArmA X for $10 more is so worth it. :)