Android - How to install a custom keyboard in Android?

Yes you can. I've used Swype as well as TouchPal Keyboard as alternatives.

Once you install any new keyboard, just go to Settings > Language & Keyboard to set your keyboard/text settings.

I found that even after changing the setting it didn't use the new keyboard. I had to do a long press in a text field. That showed an option for which input method to use. Once selecting the new keyboard it worked all the time.