A BFG is a firm favourite Bigger Stick for the Anti-Hero. If a supporting character has it, there's three possible reasons; The Big Guy will often have one as his Weapon of Choice, while other characters might pull one out or be given one if they're about to have a heroic Last Stand. A weaker character, particularly the Damsel in Distress, might happen upon the hero's discarded weapon after the aforementioned scene where the hero drops it; typically they'll struggle to aim it at the right person but ultimately succeed, upon which they'll either swear they'll never try to fire the weapon again or, more rarely, ask if they can keep it.
If a bad guy has it, expect him to be The Dragon, The Brute or a Psycho for Hire; the Big Bad will usually arm himself with something smaller but eviler unless he is an Evil Genius, in which case his BFG is likely to be a Wave Motion Gun so big it has its own seat. It might be suggested an evil user is Compensating for Something, which might well be a bad idea. If the weapon is the bad guy's gimmick, it will most likely have a name, and might get a speech where the villain gloats about how big and awesome it is.
It's often a sign that the user is literally compensating for something; a bad guy using such a weapon is quite likely to go down from one punch after being relieved of it if he isn't also a Brute. It's typically also used to escalate the action; much like we know The Brute is a threat because he's physically imposing, we can tell a BFG-equipped Dragon or Cold Sniper can do a lot of damage straight away, because their gun looks like it can. The BFG's range and firepower is likely to force the hero to defeat the user with cunning rather than straight-up firepower (much like if the bad guy has a powerful vehicle); during this standoff, if the hero is armed with something more normal, his "puny weapon" is likely to be compared to it. If it has any kind of weakness (ammunition drum, flamethrower fuel tanks, ability to overload, etc) expect the user to be hoisted by his own petard.
In older videogames, the BFG is typically the last weapon the player acquires (though there might be a very well-hidden one earlier on), and coincides with the introduction of the toughest monsters; if it's powerful enough, it might go all the way up to Eleventh-Hour Superpower, with the player fighting the game's toughest normal enemies as if they're just regular grunts; usually the ammo will be rare enough that it's restricted to only the most extreme situations, however. This can result in the weapon being Too Awesome to Use. More modern games will typically limit a BFG by either fixing it in place (the result usually being Hold the Line), using an inventory system that doesn't allow the player to keep hold of it, or just restricting ammo to specific boss areas so the player has to discard it afterward. If an enemy's got one, it usually instantly qualifies them as at least an Elite Mook; BFGs are also the favourite armaments of the Giant Mook, King Mook and Boss in Mook Clothing. Game BFGs sometimes suffer from effects related to their awesomeness, such as Overheating, and may have a Charge Meter.