What is the word for a person or people who enjoy pain?

On themself, or others? If others, I belive the word you are looking for is "sadist".

It's a masochist. Just look at this wonderful song by Tom Lehrer and you will understand.

http://www.lyricsfreak.com/t/tom+lehrer/the+masochism+tango_20138389.html

Old joke: How do you be cruel to a masochist? Don't hurt him... The question is ambiguous, though. Sadists enjoy pain - when someone else is suffering it. Be a sado-masochist and get the most out of life...

The word “algolagnia” means taking pleasure in pain, whether in inflicting it or suffering it or both. I can’t find a word for a person with this tendency: however, my OED has the adjective “algolagnistic,” so perhaps the word is “algolagnist.” “Algolagniac” might also be a permissible form.

al·go·lag·ni·a ( P ) Pronunciation Key (lg-lgn-)
n.
Sexual gratification derived from inflicting or experiencing pain.

Main Entry: al·go·lag·ni·ac
Pronunciation: -nE-"ak
Function: noun
: one who practices algolagnia

Reference.com

Beat me beat me cried masochist. No No replied the sadist.

Question #97780 (DELETED). EadsLover asks:

What is the term for people who enjoy pain. I know when people enjoy pain being inflicted on others it's called sadism, but when about enjoying pain on one's self?

sequoianoir replies:

Masochism

BRY2K replies: Someone who obtains pleasure from receiving punishment would be termed a masochist.

dj168 replies:

I would say Emo.

https://en.wikipedia.org/wiki/Emo

Both can be applicable, but they're different. A masochist enjoys pain inflicted by another, while an Emo goes more for self-inflicted injury. (Not all Emos, by the way - that's the image they have because of the music type and the activities of certain of them.) Masochism is most often associated with sexual pleasure, but there can be cases without that element being present. This is not the case with Emos as a rule. I can't think of a word that covers both categories. See above references, and I was going to post a link to the Tom Lehrer.....