The most common would be the classic Tri-Fold. it is mostly used for product pamphlets, take-out menus, promotional brochures, and many other applications. It can be easily inserted into envelopes, so it's a great option for mailing. A "Letterfold" is basically a portrait Tri-Fold. It has a different name because it is mostly used for letters and documents inserted into a #10 envelope.
A variation of the Tri-Fold is the Z-fold, where the last panel is folded outwards instead of being tucked in, forming a "Z" shape. Z-fold is mostly used for bilingual items, as each side of the folded brochure serves as a front cover panel.
Another popular fold is the half-fold, as it is simple and economical. It is good for product handouts, menus, newsletters, guides, or inserts for catalogs. Thicker paper stock can be used because there is only one fold, and additional finishes can be added without affecting the fold's functionality.