151 proof rum is one of several overproof rum strengths. 151 seems to be the most common strength, but overproof rums can range from the low 100's to 169 or higher proof. So why do we have this broad category of 151 proof rums? The best answer to that question is that the old pot stills that all rum was made from naturally produced a distillate at around 151 proof. Normally the rum would be cut with water before bottling (though back then it was probably cut to 90 to 110 proof unlike the 80 proof that we are used to today).
But if the rum wasn't cut, it remained at it's natural strength of about 151 proof. It's easier to ship high proof rum and cut it at it's destination, so we probably had 151 rum making it's way around the world in this fashion. Naturally some was sampled and found to be capable of standing up to dilution in drinks and for drinking straight in pissing matches.
There were probably all kinds of other therapeutic, medical, and snake oil uses found for 151 proof rum as well. Add to this the catchy sound of the number 151 - a blessing for marketing - and you get where we are today.
Because it is so strong, 151 proof rum is almost always used with a mixer (unless you're trying to haze someone). I have had a shot of Bacardi 151 on a couple of occasions. It is very dry, astringent, and full of unbalanced, chaotic heat. Not for the faint of heart. I much prefer it with a mixer.
Demerara 151 rum is far superior to Bacardi, and has a wonderful smokey flavor. It is used as a floater in quite a few classic Tiki drinks.
151 is also used in flaming drinks as it lights quite readily. I have also found 151 proof rum useful when you want to transport some alcohol but you have to keep the weight down, such as when you are backpacking.