How can I check if my computer's powerful enough to run a certain game?

For Windows 7 / Vista you can use the Windows Experience Index score.

Or, for a more precise estimate, you can use "Can You Run it?":

https://www.systemrequirementslab.com/

This runs as a Java Applet that will actually benchmark your system and compare it to the performance of the game.

Demos, though increasingly rare (and massive) can be a good way to verify compatibility. Just remember that a prerelease demo may be poorly optimized compared to the final game.

Most modern PC games will publish the recommended at minimum system requirements at some point in time before the game is released. For example, the game Witcher 3, released in North America on May 19th, 2015, released their system requirements in early January of 2015.

The System Requirements Lab linked by John Gietzen is an excellent resource, but if you are unable or unwilling to run that applet, the publishers' minimum requirements will be a good place to start.