Pedair Cainc Mabinogi (The Four Branches of the Mabinogi) is the oldest recorded British storytelling in prose, and appears to go back to about the eleventh century. It consists of four story sequences, each concerned with a particular character: Pwyll, Branwen, Manawydan, and Math. The sequences are themselves united by a common background -- they all take place in a legendary Britain united under various principalities but united by a single king -- and by the character of Pryderi, who appears in all four sequences. It has survived in two manuscripts -- the White Book of Rhydderch and the Red Book of Hergest, both of which were copied in the fourteenth century -- and some fragments. The name Maginobion comes from this set of story-sequences; the name itself was a passing medieval scribal error (it is a pluralization of an already plural word), but it stuck when William Owen Pughe used it at the end of the eighteenth century for his series of translations into English, "The Mabinogion, or Juvenile Amusements, being Ancient Welsh Romances".
The Mabinogion as we usually recognize it, however, is due primarily to Lady Charlotte Guest. Lady Charlotte herself had a genius for languages; she was fluent in seven languages before she even started Welsh, and she is the source of quite a few excellent and influential translations. Guest used a transcription of the Red Book of Hergest for her translation in 1838 and pulled together a number tales to make a full anthology; this is what we refer to as The Mabinogion. The tales in Guest's anthology:
Pwyll, Prince of Dyfed
Branwen, Daughter of Llyr
Manawydan, Son of Llyr
Math, Son of Mathonwy
The Dream of Macsen Wledig
Lludd and Llefelys
Culhwch and Olwen
The Dream of Rhonabwy
Owain, or The Countess of the Fountain
Peredur, Son of Erefawc
Geraint and Enid
The Tale of Taliesin
Of these, The Tale of Taliesin is sometimes included as part of The Mabinogion, and sometimes treated as if it were a distinct appendix; it is the only tale in the connection not found in either the White Book or the Red Book. (The Dream of Rhonabwy is only found in the Red Book, but all the others are found in both.) Likewise, the collection of these stories in particular is an entirely modern affair; they are not found grouped together in their original manuscripts, and have little in common beyond all being prose rather than verse. Thus, like a lot of national literary works, The Mabinogion is partly natural and partly artificial; it pulled together a number of genuine native elements into a form that is itself entirely modern. The Mabinogion itself is, in a real sense, just Lady Charlotte Guest's 1877 final edition of her translated anthology; any other is version is taking Guest's anthology as a template.
And it is Lady Charlotte Guest's Mabinogion that is the next fortnightly book. For the Four Branches of the Mabinogi, I may also look at Will Parker's online translation of the White Book text, for comparison, if I have time.