The ISO system
ISO 216 defines paper sizes with an aspect ratio of 1 : √2 (about 1 : 1.414). That ratio has a special property: cut a sheet in half across its long side and both halves have exactly the same proportions as the original. A0 is defined as one square metre; A1 is half of A0, A2 half of A1, and so on down to A4 (210 x 297 mm), the world's standard office sheet, and A5, common for notebooks and books.
The B series fills the gaps between A sizes (B5 is between A5 and A4) and is used for posters, passports and some books. The C series defines envelopes: a C4 envelope fits an unfolded A4 sheet, C5 an A4 folded once, and the long DL envelope holds A4 folded in thirds.
North American sizes
The United States and Canada use inch-based sizes: Letter (8.5 x 11 in), Legal (8.5 x 14 in) and Tabloid (11 x 17 in). They do not share a single aspect ratio, so scaling between them crops or leaves margins. Technical drawings use the ANSI sizes (ANSI A is Letter), and architects use the Arch series.
A4 vs Letter
A4 is 6 mm narrower and 18 mm taller than Letter. Documents designed for one and printed on the other usually shrink to fit or lose a strip at the bottom. If you share files internationally, leave generous margins or produce both versions.
Japan and other variants
Japan has its own JIS B series, slightly larger than ISO B. JIS B5 and B6 are common for notebooks, magazines and manga. Many countries also kept traditional sizes — see British imperial sizes and East Asian formats.
Paper sizes and book trims
Book trim sizes are not paper sizes, but they are related. The digest trim (5.5 x 8.5 in) is half a Letter sheet; A5 books are half an A4 sheet; and the British Demy and Royal book formats take their names from traditional sheets. Printers choose press sheets that a book's pages fit onto efficiently — one reason some trims are cheaper than others.