The shotgun is devastating at close range, but quickly tails off in performance after that. It fires a spread of pellets, which will leave individual marks in a wall if you want to count them. You only get to use this in the second half of the game, when things start running at you gurgling. Take every spare moment to reload, even if it's only a few shells. Best to be prepared for the worst!
Being keener on precision weapons and safety-based play that involves keeping some distance from the enemy where possible, I don't lust after the shotgun as much as some players do, but I'll admit that I do find it highly satisfying to blast Elites at point blank range! That really works things out of your system eh? Payback time. But I don't have much to say about the weapon tactically. I'm not even sure there is much to say. I mean, you get mighty close and pull the trigger, and BLAM! you take out the enemy and a large portion of the nearby scenery. That's about it really. Brutal.
Exploiting the spread
However, here's something. When faced with a group of enemies close together, a situation you often get with the Flood, you can potentially use the spread of the blast to stun and damage multiple enemies at once. It may not kill them, but it at least stops them shooting for a moment. You can then follow up, possibly using further 'shared shots' until someone goes down, and hopefully finish everyone off before taking any damage at all. This method can be preferable to focusing on one enemy at a time, since others could be busy shooting you in the meanwhile.
Turn and shoot
Here's something else now I come to think about it (having lately been using the shotgun a lot more than I used to). The shotgun is easily the weapon of choice if you like wading in amongst enemies and dishing out hurt at close quarters. With enemies closing in on multiple sides - Flood in particular - it can be a very enjoyable case of repeatedly turning and shooting to take them down one after another, all the while keeping evasively on the move of course. In such a situation it really helps to be playing with headphones or (I imagine) a sound-surround system, as you'll be able to sense the threats around you just by sound. Picking out a close sound, you can potentially turn and shoot in one go, taking down a target before even sighting him. This coordinated 'turn and shoot' move is something you can become better at with practice. It can be used with other weapons too of course, but they won't have the impact of the shotty, plus you'd need to be more accurate with your aim.