Le "demandeur" ou "le requérant", est la personne physique ou morale qui a pris l'initiative d'engager une procédure judiciaire en vue de faire reconnaître un droit. Son adversaire est le "défendeur ".

But not applicant
Claimant is now the term used in civil proceedings in England & Wales as well as many Commonwealth countries
Plaintiff is still in widespread use in many countries including the US
Plaignant and demandeur have the same meaning in civil cases, but I have never seen plaignant used in this sense. It is more commonly used in criminal matters where the nearest English term would be "complainant"

Complainant - see Allegro's explanation, besides being used for complaints to the UK police, is also used in tribunal cases in E+W e.g. for employment and for US civil complaints.

demandeur is Scots law would be called a pursuer and the opposing party a defender.

The supplicant of a civil injunction in E+W is described in High Court writs in the alternative as 'applicant / claimant' and the opponent as 'respondent / defendant', the reason being that the titles of the parties change on the prelim. case turning into a fully blown action.

Middling confidence level as UK Solicitors - as traditionally 'difficult' translation clients in London - used to 'complain' pre-Lord Woolf civil justice reforms in 1998 - unwisely about the > into ENG translation by an inhouse Litigation Solicitor colleague of mine - that the correct term for a civil restraining injunction is a 'claim' and not an 'application' for such process.

Otherwise, I agree with with the Discussion Entries of the Discussion Intrants laurent Di R. and Schtroumpf.