A Monster Manual (sometimes called a Monstrous Manual or a Bestiary) is a book (or set of pages you can put in a binder if you are a particularly crusty neckbeard) used in RPGs to describe the various kinds of monsters the PCs can encounter and fight. If they are not bundled into the main manual, a new release of a game can see a new Monster Manual as well. This first Monster Manual is seen as a "core book" in the trinity of the Manual, the Handbook and the holy Guide. The Manual is intended for DMs to make encounters for the players. Though they are more often described in the DMG, these books can also contain descriptions for the more exotic kind of trap (no not like that). Some books may also include templates to apply to existing monsters to change them (read: make them deadlier) or ways to make your very own monster from the ground up.
After the first Monster Manual of an edition the follow-up books are divided into two different camps. One is the list of Monster Manuals who will be named Monster Manual II, Monster Manual III and so on. The other are the more "themed" books that describe settings for adventures and monsters that fit in those settings, like books describing the Underdark having many Drow, spider and Aberration type enemies, or The Manual of the Planes describing Fiends like Tanar'ri and Baatezu, alongside creatures of Chaos like the Githzerai and creatures of Law like Modrons. Books in the latter category are not considered Monster Manuals despite their number of described creatures.
The problem with making a lot of Monster Manuals is that the monsters in the later books are split up into four groups: 30% will be reprints of monsters from older books of varying obscurity, 30% will be either upscaled animals or creatures made by slapping a number of templates together ending with creatures that lack the focus and originality of their progenitors, 30% will be hilariously stupid creatures that no serious DM would ever use (Killer walls! Killer paper! Murderous hats!), and the final 10% being actually interesting creatures (either original or mythological) that could make an interesting addition to a game.
Theoretically the game can be played without the Monster Manual and just the PHB+DMG by making humans and traps the only enemies. In practice this is never done for long and the Monster Manual is an important part of any game, as a game with nothing interesting to fight is just as bad as uninteresting mechanics or classes.