What is the name of the forest that borders the Shire to the east?

The forest bordering the Shire to the east is the Old Forest.

The Old Forest is an ancient, unsettling woodland located just beyond the Shire's eastern boundary, across the Brandywine River. Unlike the carefully cultivated countryside of the Hobbits, its trees seem hostile and strangely aware. The forest is a remnant of the vast woods that once covered much of Eriador.

Frodo, Merry, Pippin, and Sam enter it during their journey to Bucklebury Ferry. They quickly discover that the paths appear to shift and that the trees seem to influence travellers' movements. Their route draws them toward the valley of the Withywindle and Old Man Willow, one of the forest's most dangerous inhabitants.

The Old Forest is often confused with Mirkwood or Fangorn Forest. Mirkwood lies far to the east in Wilderland, while Fangorn stands near Isengard. The Old Forest is geographically tied to the Shire, the Brandywine, the Withywindle, and the Barrow-downs, making it part of the Hobbits' immediate western-Eriador landscape.