The most widely recognised music examining body worldwide is The Associated Board of the Royal Schools of Music: ABRSM. Although based in the UK, they operate worldwide.
There are a number of other examining bodies, including:
- Trinity Guildhall
- London College of Music Examinations
- Victoria College of Music
- The National College of Music London
- Australian Music Examinations Board
- New Zealand Music Examinations Board
- The Royal Conservatory of Music
- University of South Africa - Directorate Music
- Vienna Music Examination Board (Wiener Musik-Prüfungskommission)
Many (but not all) boards follow the same structure and approximate standards as ABRSM, and it is ABRSM exams which I describe here.
There are 8 grades for theory and 8 grades for practical (that is, performance). Most people alternate between theory and practice exams as they work up through the grades.
Theory exams are entirely written. Practical exams involve performing in front of an examiner, and (as an illustration) consist of exercises, sight reading, and performing a practised piece.
It is not the case that Grade 8 qualifies you as a professional musician.
Many music colleges require Grade 8 as a minimum entrance requirement; by that logic Grade 8 is only the beginning of your musical study.
Many professional jazz/folk/rock/pop musicians have no exam qualifications whatsoever.
ABRSM syllabuses are available online.
I don't believe there is any formal arrangement whereby examining bodies recognise each other's qualifications, but it doesn't really matter since someone who thought they had the skill level could jump straight in and do a Grade 8 exam without doing the preceding grades, as long as you pay the examination fee.