From the northern mexican perspective it's really just about wrapping something in a flour tortilla. Im sure people are going to post a couple of nice tex mex, southwest or New cuisine wraps so I'll go ahead and post the totally north mexican burritos:
Burrito de machaca de res: For this type you need to rehidrate dehadrated, salted shredded beef. Otherwise use reagular pulled beef and give it a good ammound of salt and spice. Fry with potato cubes, little bits of tomato, one diced jalapeno or serrano and chopped onion and you are set. All you need to do is wrap this into thin (about 1 inch thick max) burritos. These are a "packed lunch" meal... something moms will make a bunch of for their kid on a field trip (a tupperware full or something) or sometimes old ladies or their kids will get on busses to sell these at stations (they will have a coleman cooler fillled with little bags of them, still steaming).
Burrito de chorizo y papa: the ultimate "Im spoiling you to death" food, usually made by some ones nanny, mom or wife when trying to pamper you. Its similar to above with cubed potatoes and onions fried together with northern chorizo which has more in common with ground meat than the peperoni like european chorizo. If you have any doubt just fry some ground beef with paprika, chilli powder (guajillo if you've got it), vinager and salt and you should get a similar result. Now stuff some of this into a tortilla and fold it in the "burrito" style which means a big glob in the middle, fold the ends and then fold the sides so it makes a nice "envelope" of food.
More classic burrito fillings (from stands that specifically sell burritos or moms with too much time on their hands):
Carne en chile colorado: a pretty accurate way to describe this is pulled beef in enchilada sauce with diced potatoes. Great stuff, it would seem like a tex mex pulled pork in bbq sauce.
Chicharon in salsa verde: It means pork rinds in green salsa... sorta. You see chicharron refers in general to deep fried nuggets of meat (even sea food) AND pork rinds. So here you take small pieces of pork and fry them in a generous ammount of oil at a lowish temperature so that they dehidrate a little, you are looking for almost crunchy pieces of pork. Once ready you simmer them in a tomatillo sauce and this makes an AMAZING burrito.
Burrito de frijoles puercos: the most generic and probably the best burrito out there... "pork or filthy" beans and cheese burrito. Frijoles puercos are refried beans with a little chipotle sauce or diced chipotle peppers for a smoky spicyness. It also has some chilorio (fatty pulled pork in a chilli sauce) and traditionally they are fried in lard, giving them a REALLY rich taste, moreso than a straight meat dish. A dab of this thick mix on a tortilla with a big chunk of asadero cheese and a slice or two of jalapeno makes a great meal.
My english, she's not so good... I meant to say I did it with the malice of forethought.