Uninsured motorist (UM) covers you when the at-fault driver has no insurance or some amount under the legal limit (and is driving illegally). IIRC, it's border states like TX, AZ, and CA that have relatively high UM claims (and thus rates).
While it's true that there might be a higher %/# of claims in the border states, don't assume that just about everyone buys the legal minimum. My father has had 2 accidents caused by uninsured motorists that were just 2 years apart. Thankfully, they were just fender benders (and funny how the cop didn't ticket the offender like they were supposed to for not driving with proof of insurance). There could be quite a few more people out there driving w/o insurance than you realize. You simply cancel your auto insurance policy the day after obtaining your license plate from the state (or even buy insurance from some companies that specialize in short-term insurance) - and the state is never the wiser.
Underinsured motorist (UIM) covers you when the at-fault driver has a legal amount of insurance, but it's not enough to reimburse you your damages. So if you're driving in your brand new $150k Masarati, and the dude who hit you had 20k damage limits, his insurance company will give you a check for 20k, and you'd have to go through the hassle of suing him (and hope he has enough assets or whatever to not declare bankrupcy) to get the other $130k.
Since mustaschians don't drive such dumb cars, it's also about medical expenses, (if their injuy limit is 20k and you have 100k in medical expenses), but your primary health insurance will probably run secondary to his injury coverage and pay the difference. It is a hassle in getting the claims submittted because most heath insurance plans will completely deny any auto injury claims until you prove that the auto insurer has paid the limits.
2 observations on the above comments:
--Other passengers in your car: don't forget that other passengers in your car can also come into play. Who's to say that if you have a family member or friend (or two) in the car w/ you, and an uninsured/underinsured person hits you, that they might not be injured? And don't forget that health insurance covers medical bills, and maybe a little rehab - not pain and suffering, and longer/intensive rehab. And if the person at-fault has no/too little coverage, then it's up to them to sue you/your insurance company. While you might trust your mother to be honest, what about that friend-of-a-friend?
--Other drivers/things: granted, we're starting to get into the less likely incidents (but we are talking about insurance here :) ), but if an uninsured/underinsured motorist hits you, you could come out ok...but what if the motorist hits you, and your car then damages other cars/objects? That's not to say that the 3rd party wouldn't try to bilk the insurance company for debilitating back pain, etc....and if the original person at fault (who caused you to hit the 3rd party) was un/under-insured, guess who's left with the assets and policy they're going after?
The cost of UIM is usually tiny, (although not in a couple of states....I can't remember which), so it might be worth the lack of hassle.
As a comparison, here's my quote for an annual policy from 2011 from GEICO in Missouri (didn't ultimately go w/ them, but they were somewhat competitive):
250,000 person/500,000 accident coverage, $100,000 property coverage
$319 liability
$51 uninsured
$47 underinsured
