Masochists enjoy having pain inflicted on them. It might not be all kinds of pain, but only specific types. They usually match up with sadists because sadists enjoy inflicting pain.
Masochists tend to be submissive, but can be switches too. Masochists know what it feels to take a punishment do can make effective tops in a scene. Some Dominants have masochistic streaks, enjoying a certain level of pain themselves. Masochism is centred on the pleasure of pain but many masochists enjoy being controlled and dominated too.
Masochists are often bratty, to get the pain they want. The worst torture for a masochist is totally withholding punishment as they crave pain.
In day-to-day life masochists may find non kinky ways to satisfy their desires such as pushing themselves hard during exercise. Masochists often use pain to feel free from the pressures of day to day life.
It's not uncommon - especially in the kinky world - for someone to process pain as pleasure, and to crave sensations a little rougher and more intensive than are usually considered the norm. These people are often called 'masochists', and while it's important to remember that the word isn't synonymous with 'submissive' (not all masochists enjoy psychological submission, and not all subs are particularly interested in physical pain!) the two often go hand in hand and each category has traits in common with the other. Masochists are the opposite number to sadists, who enjoy inflicting consensual physical pain just as much as their masochistic partners enjoy receiving it.
There's more than one 'official' definition of what the letters in BDSM stand for, and plenty of people don't agree on their exact usage! However you use the term, though, it's a catch-all way of describing a whole bunch of kinks and fetishes that are more common than many people imagine: bondage and impact play, sadism and masochism, domination and submission.
Safety, consent and respect for boundaries are all key in the practice of BDSM, and it's vital to negotiate properly with your partners before getting into anything heavy-handed. Thankfully it's not so hard to learn - most places around the world have their own BDSM communities that are only too happy to take in newbies and help them learn the ropes.