A tiling window manager arranges your computer's windows automatically. Instead of floating windows that overlap, each window gets a part of the screen. No more manual resizing, dragging windows around, or even minimizing! dragging edges, or even minimizing!!
Why Use One
Everything is keyboard driven. You open, close, move, and resize windows without touching the mouse.
Tiling window managers also lean heavily on virtual desktops, usually called workspaces. Each workspace holds its own set of windows. You arrange these as needed and switch between them with a keypress. This keeps each workspace focused instead of stacking every window into one cluttered space.
If you have multiple windows on the same desktop they can arrange themselves in different ways depending on the window manager. Some use fixed splits, dividing the screen into a grid as you open new windows. Others use binary space partitioning, splitting the space in half each time a window opens. Other systems exist but are less common.
The Options
i3, and its Wayland counterpart sway, are the default choice for most people getting started. They use a fixed-split model and a plain text config file, which makes them easy to configure and easy to get help with. I used i3 for five years, across both a laptop and a desktop originally with a plan to "graduate" to a more complex window manager but realized there was no real reason to do so.
bspwm is a popular alternative. It uses binary space partitioning, which gives it a more dynamic, tree-based layout than i3's fixed splits. It's also scripted through a separate program, sxhkd, for keybinds, and configured through shell scripts rather than a single config file. This makes it more flexible, at the cost of a steeper setup.
dwm exists for people who want to go further. It's part of the suckless project, written in C. There's no config file. If you want to change something, you patch the source and rebuild.
Across all of these, keybinds are fully custom. You decide what opens a terminal, what moves a window to another workspace, what closes a window. This is part of what makes tiling window managers feel personal. No two setups look the same.
Not a Desktop Environment
A tiling window manager doesn't come with a status bar, an application launcher, a notification daemon, or a wallpaper tool the way a desktop environment does. You choose your own stack. For a status bar, that might be i3bar, which ships with i3 and covers the basics, or polybar, which takes more setup but supports more modules and more customization. For an application launcher, that might be dmenu, the minimal suckless launcher, or rofi, which adds more features like themes and file browsing on top of the same idea.
This allows you to swap any piece of your setup out without touching the rest of the system.
Ricing
While I primarily like tiling window managers for productivity reasons, I'd be lying if I didn't enjoy tweaking the look of my setup too. Since every piece is separate, every piece is also themeable. Colors, fonts, gaps between windows, borders, the bar, the launcher: all of it is a config change away. Communities like r/unixporn exist because of this. People share screenshots of their setups, and the config files behind them, so you can see exactly how a look was built and steal the parts you want.