ALCM Associate of the London College of Music
ATCL Associate of the London College of Music
ARCM Associate of Trinity College London
ADMT Associate Diploma in Music Teaching
Dip ABRSM Diploma Associated Board of the Royal Schools of Music
Then there are Licentiate qualifications, which are an extension to the Associate qualifications:
LRSM Licentiate of the Royal Schools of Music
LLCM Licentiate of the London College of Music
LTCL Licentiate of Trinity College London
LRAM Licentiate of the Royal Academy of Music
LRCM Licentiate of the Royal College of Music
LGSM Licentiate of the Guildhall School of Music
A Fellowship follows the Licentiate qualification, for example FTCL is a Fellow of Trinity College of Music London. You may come across other qualifications such as a Diploma in Music, which is usually abbreviated to Dip.Mus, or a Graduate Diploma, abbreviated as Grad.Dip.Mus.
The Bachelor of Music is abbreviated as B.Mus.
Of course having letters after a name doesn't necessarily mean he/she is a good teacher. However, it does show that they have been through some rigorous formal training and you would be safe to assume that they have reached a high level on their given instrument along with gaining a vast knowledge of music as a whole.
Now all you have to do is find out if they are any good at imparting that knowledge to others!
If you are looking for a new teacher I recommend that you ask for a trial lesson or two. This will give you and your child the opportunity to meet the teacher without getting tied into any long-term commitment.
Try to establish their teaching style to see if it will suit your child.
For example, does the teacher simply push forward with exam after exam or do they include other music and new skills between exams?
What are their weekly practice expectations? Will your child be able to meet those expectations?
Is the teacher a performer? Will he/she be away alot? Will they be able to make up lessons?
Do they teach during school holidays as well as term time?
Will the teacher also cover some basic music theory? Do avoid the quick 10mins theory tacked onto the end of a lesson. This isn't long enough to learn anything new or explain anything fully.