Fredegar Bolger was one of the Conspirators, a small group of Hobbits who knew that Frodo had the Ring. Fatty was a descendant of Hildibrand Took (T.A. 2849–2934), one of the many sons of the Old Took. He was the son of Odovacar Bolger and Rosamunda Took.
When Frodo Baggins, Sam, Merry and Pippin set out to take the Ring to Rivendell, Fredegar stayed behind in an attempt to keep up appearances and delay news of their departure. He was frightened half out of his wits by the arrival of the Nazgûl but escaped unharmed. Fredegar could have gone with Frodo and his companions into the Old Forest, but was terrified of the stories about it. Though Merry tried to persuade him that the Old Forest would be nothing compared to meeting the Ringwraiths, Fatty was adamant, so the other Hobbits went into the forest with only the knowledge of Merry to aid them.
During Saruman's rule of the Shire he led a group of partisans fighting against the Ruffians around the hills of Scary, before eventually being captured. He was imprisoned and starved, meaning that after his release nobody could call him Fatty any more.
In earlier drafts of The Lord of the Rings, Fatty Bolger played a much larger role, but this was later abandoned. His role survives as a minor anomaly in the book — before the Hobbits go into the Old Forest, they have five ponies prepared — but only take four. Bolger was originally supposed to go with them.