You're not missing much if you skip the original Bad Company. Stick to Modern Warfare instead.
Battlefield is a series made famous by it's vehicular online battles. Notice that I didn't say that it's a game famous for single player.... Well, remember that: it'll be important later.
Bad Company is a spinoff of the traditional Battlefield series, created by DICE to give themselves the opportunity to put more thought and effort into the single player campaign. Sadly, the single player campaign is definitely an improvement over previous entries, but it's still a long way from fun.
You play as Preston Marlow, a newbie joining "B Company," nicknamed "Bad Company" since it's made up primarily of troublemakers, idiots, and outcasts. Your job is simply to follow your sergeant's orders for the game. Not hard. But the game itself does run into a few issues.
First and foremost, the very biggest problem: you can only carry a single gun at a time. Instead of having two guns (like almost every other shooter these days), Bad Company gives you a single gun with multiple firing modes (except with sniper rifles, then you get a pistol as well). Guns that don't have a second firing mode allow you to manually throw grenades.
This is a problem. The two-gun style started, at least in my mind, with Halo (or at least, Halo was my first experience with it). Its purpose seemed simple: force the player to use less weapons, thereby making him think and decide about what two to bring. It added a small element of strategy to the game and made it feel more tactful. Being able to carry only one gun is just silly. There is no "tactics" involved at all; I used the standard gun I started the level with for almost the entirety of the game with no problems at all.
On the other hand, Bad Company lets you use various gadgets, and to great effect. You always have your auto-injector gadget. Unlike most other shooters, you don't automatically regain health; you have a health meter. But you can use the auto-injector to heal yourself (infinitely). It makes the game much more tense, as you have to keep a fairly close eye on your health and be ready to whip out that auto-injector. Other gadgets include a laser designator (for calling in airstrikes), a power drill (for repairing vehicles), and a rocket launcher (for disrepairing vehicles).
A couple of other points in its favor is some minorly clever writing. It won't win a Newbery Medal or anything, but it does manage to put a fresh spin on tired, largely archetypal characters. The voice acting (and actually, overall sound design in general) is also very sharp. Characters sound nice, gunshots are loud and grumpy, and explosions are deafeningly impressive.
Perhaps Bad Company's biggest claim to fame though is its destructability. Almost everything in the game can be destroyed. You can destroy buildings, turrets, vehicles, people, anything. This presents certain gameplay options that are not available in other games. For instance, in other games, if you see an enemy soldier retreat up the stairs of a house, you follow him. In Bad Company, you don't have to. If you'd rather, you can go outside, get in the nearest tank, and simply annihilate the top half of the house. Unnecessary... yes, but damned funny!
Watching stuff blow up doesn't seem to get less fun as the game goes on. So how could a game based largely on blowing stuff up possibly be not fun?
Let's discuss vehicle controls. Battlefield is, ironically, famous for its vehicles. We know this is true. So why do all the vehicles in Bad Company control in a manner I can only really define as "slippery?" Remember trying to play Oblivion in third person? It's kind of like that, only not quite as bad.
Also, Bad Company is a war game. You are a soldier. Then why is it everytime you get to a difficult part with a lot of enemies does simply "legging it" seem to get the best results? I was able to win a fight via flight much more often than standing my ground and duking it out.
And lastly, the new checkpoint system is rubbish. When you die, rather than going back to the checkpoint (i.e., restarting the section of the level you're on, losing all progress since the last checkpoint), Bad Company just has you go back to the physical location of your last checkpoint. But unlike most other games, all the enemies you killed are still dead, you have the same gun you had at death (with the same amount of ammo, which can REALLY suck if that ammo count is zero), and thus, you haven't really lost progress. This system sounds good on paper, but it can be equally if not more annoying than a standard checkpoint system. The spots where you get respawned are often very far away from where you were, and you're left to run a good 2 to 3 minutes back to the battle. And I suppose at this point I should mention that your character doesn't "run" so much as "saunters."
And the final boss battle was the annoying cherry on the annoying sundae. To avoid giving anything away, let's just say that a few more hit maps would've helped a lot. It's annoying enough that it completely sucked away the thrill of victory for me.
The game clocks in at a somewhat measly seven hours on the normal difficulty, which I suppose I should be thankful for since I didn't care much for it. All in all, you're not missing much if you skip Bad Company. Stick to Modern Warfare instead.