Windows-Logo-Key + Left/Right Arrow -- Keyboard shortcut equivalent on Mac

BetterTouchTool lets you set up custom keyboard shortcuts to snap windows to the left/right/top/bottom (among other things). You can also create custom gestures, and per-app gestures/shortcuts.

If you hold down the green fullscreen button, it gives you the option to split/share the screen with a second window

Since asking this question many years ago, I eventually stumbled onto Spectacle https://www.spectacleapp.com/. Comes with window management out of the box with similar keyboard shortcuts (not exact, but actually more expressive than those built into Windows)

Highly recommend this for people looking to do window management on a Mac the same way you can on Windows.

I'm personally using ShiftIt to accomplish this task which allows many keyboard shortcuts - among them is the ability to do this with Ctrl+Option+Command+Left/Right