A collection of Welsh tales first preserved in the early Peniarth manuscripts (ca 1225), then the White Book of Rhydderch (Llyfyr Gwyn Rhydderch) written about 1300-1325; and later in the Red Book of Hergest (Llyfyr Coch o Hergest), written between 1375-1400. Most scholars believe that all but the three "romances" date to the tenth or eleventh century, and are based on much early mythology. 

The name is a misnomer, as the word "mabinogion" doesn't exist in Welsh. It was a mistake made by the scribe, existing only at the end of Pwyll pendeuc Dyfed. The real word is "mabinogi," which some translate as "tales for youth," "tales of the Mabon" and "tales of the hero." They derive this meaning from "mabon"--meaning a young man or youth. It is also the name of a god, Mabon ap Modron. This name only applies to the first four tales. 

The tales can be divided into four sections (five if one includes the story of Taliesin): 

The Four Branches of the Mabinogi: detailing the exploits of the Children of Llyr and the Children of Don; more specifically, it chronicles the life of the hero Pryderi ap Pwyll. This section is the most clearly mythological, with much material analogous to the ancient Irish sags: 
Pwyll penduc Dyfed: Pwyll Lord of Dyfed 
Branwen uerch Llyr: Branwen daughter of Llyr 
Manawyddan fab Llyr: Manawyddan son of Llyr 
Math fab Mathonwy: Math son of Mathonwy. 
The Histories: These two stories deal with early British history; they are not unlike episodes out of Geoffrey of Monmouth's History of the Kings of Britian: 

Breuddwyd Macsen: The Dream of Maxen 
Cyfranc Lludd a Llefelys: Lludd and Llefelys 
The Tales of Llys Arthur: These are tales generally only found in Wales; they have no analogues on the continent, though Culhwch ac Olwen is reminiscent of the story of Jason and the Argonauts. 

Culhwch ac Olwen: How Culhwch won Olwen 
Breudwyt Ronabwy: The Dream of Rhonabwy (found only in the Red Book) 
The Franco-Cymric Romances: These have obviously been influenced by the French romances of Chretien de Troyes; still, they contain some native elements, and are not simply derived from Chretien, but influenced by: 

Owein; Chwedyl Iarlles y Ffynnaw: Owain, or the Lady of the Fountain--analogous to Chretien's Yvain. 
Historia Peredur fab Efrawg: Peredur son of York--analogous to Perceval. 
Gereint ac Enid: Gereint and Enid--analogous to Eric et Enid.
When Lady Charlotte Guest translated the books, she added the story Hanes Taliesin; however, the manuscript from which that story comes is quite late in date, possibly the sixteenth or seventeenth centuries. It is comprised of two parts--the Story of Gwion Bach, and the Story of Taliesin, both of which are sometimes found without the other. Lady Guest's version was derived from Iolo Morgannwg's copy, while Ford's copy (see below) is from Elis Gruffudd's 16th century Chronicle of the World. 

The order of the tales in the White Book are as follows: Pwyll, Branwen, Manawyddan, Math, Peredur, Maxen, Lludd, Owein, Gereint, and Kulhwch. (The section containing Rhonabwy is missing, though J. G. Evans believed it had likely sat between Lludd and Owein.) The order in the Red Book is: Rhonabwy, Owein, Peredur, Macsen, Lludd, Pwyll, Branwen, Manawyddan, Math, Gereint, and Culhwch. Our current order: Pwyll, Branwen, Manawyddan, Math, Lludd, Maxen, Culhwch, Rhonabwy, Owein, Peredur, and Gereint, derives from J. G. Evans' diplomatic version of the Red Book Mabinogion, in the late 19th century. 

There have been five translations into English: 

1839--1849: translated by Lady Charlotte Guest in seven volumes. This is the only version which features Taliesin. It now exists as a Dover edition, which is unfortunate, since they don't print her end notes, which are full of folklore not recorded elsewhere (at least not in any form available to the average 21st-century hobbiest). Otherwise, if you can find a copy of the 1902 J.M.Dent/Everyman edition, buy that. (I got mine in London for £5. I was ecstatic.) There is--published only in Britain--a new version with illustrations by Alan Lee, IIRC. However, it isn't published in America. It was originally published in three volumes: I. Owain; Peredur; Rhonabwy; II. Geraint; Kilhwch; Maxen; III. The Mabinogi; Lludd; Taliesin. 

1929: The Mabinogion. T.P. Ellis & John Lloyd. Oxford. Published in two volumes: I. The Mabinogi; Macsen; Kulhwch & Olwen; II. Rhonabwy; Owein & Lunet; Peredur; Gereint & Enid. 

1948: translated by Gwyn Jones and Thomas Jones. This is the most accurate translation--that is, word-for-word--but because of this it is somewhat unreadable. Published by Everyman. 

1976: translated by Jeffrey Gantz. This is the most readable translation, written in modern English but still fairly true to the text. Published by Penguin. 

1986: translated by Patrick K. Ford. This is now out-of-print. It doesn't feature the Romances, but does feature Taliesin. Published by University of Calfornia.