How to stack open windows side by side

Horizontally:

Just drag a window to the screen edge and it should automatically take up half of the screen. Do the same for the other edge and you should have two equal sized window tiles next to each other.

You can semi-maximize current window to the left or right side of the screen by pressing Ctrl + Super (Windows key) + Left or Right. Press and hold the Super Key to see all available Keyboard shortcuts.

Since this is 8 years old now, an update to the answer by thanos for Ubuntu 20:

Super (Windows key) + Left Arrow or Right Arrow

will put a window half-maximized to the left or right half of the screen (Ctrl key not required anymore).