Idylls are short poems about rural living, but Theocritus’s original idylls were quite long by the standards of modern poetry. This is generally seen as a synonym for pastoral. When the two are differentiated, it is accepted that “idyll” will then refer to poems directly inspired by Theocritus’ writing.
Bucolic poems, depending on perspective, are either a subcategory of pastoral poems OR a synonym for pastoral. (It’s a bit of a gray area.) The most noteworthy examples of bucolic verse tend to idealize rural living as an aesthetic rather than as a lifestyle, with more appeal for the upper class.
Pastoral poems are known for romanticizing idyllic country life and exploring man’s relationship with nature. A pastoral poem expresses one’s desire or fantasy to enjoy a more peaceful life in the countryside.