There are far too many rules to explain in one post,but I'll give you the basics. There are three types of mutation Meddal(soft), Trwynol(Nasal) and Llaes(Aspirate). The letters that mutate are, T, C, P, D, G, B, Rh, M and Ll. Of those all nine mutate softly (except when it's a weak soft mutation, but don't worry about that for now since it only means that Ll and Rh don't mutate in those situations. The first six (T, C, P, D, G and B) can mutate nasally. Finally only T, C, and P mutate aspirately. (Here is a link to a grid showing what I just told you: http://38.media.tumblr.com/fe5b9ee6b801fa447a108380e7a6b75b/tumblr_inline_n75yldhZjI1rs95yq.png).
Next I will show you one rule for each type of mutation, but there are many more for each type, but don't worry once you get the most common situations down you've pretty much got mutations covered for the majority of the time.
Firstly, a soft mutation occurs after the possessives "dy" (meaning your) and "ei" (when it means his, take note of this since "ei" can also mean her, but then it causes another type of mutation. So for example if we take the words "Cath" (Cat) and "Potel" (Bottle) then put them with the possessives they become "dy/ei gath" and "dy/ei botel". (Another example of soft mutation is feminine nouns after "y/yr/'r" which is what has happened in the sentence above, but note that Tafarn is one of the few words that can be either masculine or feminine so "Y tafarn" is just as correct as "Y dafarn", the feminine nouns mutation softly after "y/yr/'r" is an example of weak soft mutation e.g Rhaw (Spade) stays as "Y rhaw" even though it is feminine.
Secondly there is a nasal mutation after the possessive "fy" (my) and after "yn" (when it means in, but yn has a few meanings and for some of those meanings it causes a soft mutation). So taking again the previous word of Cath and Potel, you have "Fy nghath" (My cat) and "Fy mhotel" (My bottle).
Finally is aspirate mutations. This occurs after the words "a" (and) and ei (when it means her). So Taking the words Cath and Potel again we get "Cath a photel" (a cat and a bottle) and "ei chath" (Her cat).
I'd recommend you learn the table and some rules for mutations since they appear in most sentences and some of the things that trip learners up the most.
(P.S one way of remembering the letters that mutate is to read out "T, C, P, Doctors Go Bananas, Rhian Marries Llewelyn")
Mutation is a defining characteristic of the Celtic language family, just like how strong verbs are a defining characteristic of the Germanic language family (i.e. irregular stem changes in the past tense and past-participial forms, e.g. swim->swam->have swum; see->saw->have seen instead of swim->swimmed->have swimmed and see->seed->have seed) If you want to learn Celtic languages be prepared to learn mutations.