A masochist takes pleasure, often sexual, in receiving pain or humiliation. A sadist takes pleasure in giving it. Put them together and you get sadomasochism, which is the S and M in BDSM, and in practice the two describe roles people step into by agreement rather than fixed types of person.
Where the words came from
Both are named after writers. Sadist comes from the Marquis de Sade, an eighteenth century French aristocrat who wrote at length about cruelty. Masochist comes from Leopold von Sacher-Masoch, a nineteenth century Austrian novelist whose best known book is about a man who asks to be dominated. A psychiatrist, Richard von Krafft-Ebing, coined both terms from their names in the 1880s.
Kink and diagnosis are not the same thing
This is the distinction that matters. Sexual sadism disorder and sexual masochism disorder exist in the diagnostic manuals, but they require distress, impairment, or in the sadism case acting on urges with a person who has not consented. Two adults enjoying agreed pain between them meet none of that. Liking this is not a disorder, and the manuals are explicit about it.
What it looks like in practice
Usually far milder than the words suggest. Spanking, scratching, biting, pinching, hair pulling, wax, and impact play with hands or implements. The intensity is negotiated, and most of it sits well below the threshold where an injury is even possible.
The part outsiders miss is who is in charge. It looks like the person giving pain has the power. In practice the person receiving it sets the limits, says when to stop, and the whole thing runs at their pace.
The word in the middle
Switch. Someone who takes either role depending on the partner, the mood or the day. Plenty of people find they prefer one side and discover the other by accident. Dom, sub and switch covers the power exchange version of the same idea.
Where the law draws its line
In England and Wales, consent is not a defence to causing actual bodily harm for sexual gratification. That is settled law and was put on a statutory footing by the Domestic Abuse Act 2021. Agreement between adults covers a great deal, and it does not cover injury. The UK law guide covers the wider position.
Booking it in London
Escorts who offer domination will state what they do and at what intensity, and those limits are fixed rather than negotiable on the night. Say what you are after plainly. Fetish and domination escorts covers arranging it, and the safe word guide covers stopping.