How to install an APK file on an Android phone?

Simply, you use ADB, as follows:

adb install <path to apk>

Also see the section Installing an Application in Android Debug Bridge.

I quote Hello Android because I can't say it better ;-)

You need to enable USB debugging on the phone itself (by starting the Settings application and selecting Applications > Development > USB Debugging), install the Android USB device driver if you haven’t already (Windows only), and then plug the phone into your computer using the USB cable that came with the phone.

Close the emulator window if it’s already open. As long as the phone is plugged in, Eclipse will load and run applications on the phone instead. You need to right-click the project and select Run As > Android Application.

If you have access to a Gmail account on the phone then an easy way (in terms of minimal set up effort) is to mail the .apk file to that Gmail account.

If you then access that account from the native Gmail app on the phone it recognises that the attachment is an app and offers an "Install" button.

As per other responses this approach also requires that you have selected USB debugging on the device.

Try this - it is remarkably easy ;-)