There is no one central admissions system for postgraduate courses. Applications are usually made directly to each individual university. For some vocational courses, eg teaching, law, medicine, social work and clinical psychology, applications are made to specific admissions bodies. See Careers A-Z for details of these.
Most courses do not have official closing dates. Exceptions include medicine, some law courses and teacher training. Popular courses fill quickly - get your application in as early as possible, ideally before Christmas or early in the New Year of your final year.
Speak to academics in your school: they may be able to suggest related areas of study and institutions specialising in your chosen field.
Consider potential supervisors carefully, and make contact with them ahead of your formal application. They will be pleased to hear from students who show a genuine interest in their research, and may be able to help you to source funding opportunities.
If you have a unique research area in mind, you may need to write a research proposal with your potential supervisor and submit it to funding bodies.