The Unity website does have a list of games made with Unity
Just to clarify, the Ubuntu Unity (the default user interface) is different from the Unity game engine (Unity3D). This post refers to the game engine.
There are also many other game engines that games for Ubuntu are made in, so the games you play don't have to be Unity based. Games made for Ubuntu can be found in the Ubuntu Software Centre
Also, not all games will run on Ubuntu but there are community projects dedicated to allowing Windows applications to run on Ubuntu.
The easiest and most popular method to do so is to use an application called PlayOnLinux. Follow the instructions to install that application, and it will guide you through getting the application running. Don't expect applications to run as well as they did on Windows. There will likely be performance drops in the application as it is trying to simulate another OS while to program is running.
PlayOnLinux can also run other applications like Microsoft Office 2007 (Not 2010 yet). If you are feeling brave, it also allows you to add a custom application that is not yet in the database. This may take some trial and error playing with the settings until it runs well.
Also, see the following question. It will help:
Where can I find websites that list games for Ubuntu? [closed]