1. Request an aisle seat. Now, I know it's largely a matter of personal preference whether you choose and aisle, window, or middle. I usually prefer window seats - but on a 13 hour flight from LA to Auckland, I was eternally grateful for my aisle seat. You automatically get more leg room (you've got the aisle to stick your legs into, but be careful not to trip anyone), and I think leg room is one of the biggest factors in airplane comfort. You may argue that the window seat has somewhere to lean against, but that never works for me - it's always an unbearably uncomfortable angle. But, if you really prefer the window to lean on, take that. Whatever you do, try to avoid the middle.
2. Try to put most of your carry-on in the overhead compartments. This is also about leg room - the less stuff you have crammed under the seat in front of you, the more space you have for your legs. Try to just keep essentials like a sweater, music player, drinks and snacks in the seat pocket and stow everything else above. To help make boarding less awkward, sort things beforehand, while waiting at the gate, if you've got time.
3. Don't be afraid to get up when you need to. On many long-haul flights, they keep the seat belt sign on for most of the time. However, as long as you're not currently in turbulence, taking off, or landing, you should get up if you need to. If you need the bathroom, to stretch your legs, or to go get some water, do it. You may inconvenience others if you're not on the aisle, but it's definitely worth it. You can go to the back and ask for water, if you want.
4. Keep hydrated. This is essential! Drink plenty of water - you dry out a lot more in that recycled air than you normally would. Avoid alcohol and caffeine if you can, but if you must have it, have it in moderation and also have some water.
5. Stretch your legs. It's important to get up and walk around every once in a while. If it's a shorter flight, this may not be necessary, but anything over about 5 hours you're really going to want to do this. At the very least, do some leg exercises. There are serious health risks associated with sitting for long periods of time.
6. Eat something. Hunger is another source of discomfort on long haul flights. On short flights, you can get away with not eating, but if you're on a plane for 12 hours, you're going to get hungry. Eat the plane food if you can (maybe you'll be lucky and it'll be good! Air New Zealand at least has pretty tasty food), and get a few little snacks at the airport (or pack them if they can be brought through security - check!).
7. Entertain yourself. Boredom makes the flight go so much slower. If you're on a transoceanic flight, chances are there will be an entertainment player in the seat in front of you. Use it! If there isn't one, watch the movie provided. Better yet, pack a few things to do - music players, books, magazines, puzzles, and portable game systems are all great ideas.
8. Sleep if you can. Another way to make your flight go faster is to sleep through part (or all) of it. Planes aren't very comfortable, so this can be quite difficult. However, if you know the best way to get yourself to sleep, you should use it. Eye masks can be helpful, as can night time tylenol and other medication (definitely use carefully as labeled!), and if you're of age and they're serving dinner, a glass of wine can help. Remember my earlier suggestion - stay hydrated! Alcohol is definitely not the best method of helping yourself sleep (but can work).
I hope that these tips help a little in making your long flight a little better. These were all written assuming a passenger in coach, with the least personal space. Of course, they're not comprehensive - share your tips in the comments!