Why do my photos look fantastic on my HTC One X, but not that good on a PC? The photos I take with my HTC One X look excellent on the phone screen, but when I copy them to a PC or laptop and view them on a larger screen, they don't look nearly as good.

Is HTC employing a filter or enhancement to make the photos look better? Could I do the same to the photos after copying them to my laptop?
First, HTC is not ...employing a filter or enhancement to make the photos look better....

Second, 
This is the color settings of the screens. If you transfer a photo taken from a Sony device camera to a Samsung mobile device, you will see a huge difference. That difference is that you can't see the vividness and cleanliness (i.e. the beauty ) you saw in the photo of the Samsung phone.

Explanation: The mobile phones has a PPI or DPI more than 300 whereas the computer monitors have around 150 - 200. Since you are seeing the pictures in a lesser pixel density while you are viewing them in your PC, you may fell that they don't look nearly as good.

If you want the beauty you saw in your phone, use Photoshop or any other photo editing app to increase the colors a little and adjust brightness and/or contrast or curves. Our fellow SE members of Photography Stack Exchange will be glad to help you.