In the Security/Users folder in my database, I have a bunch of security groups, include \"MyApplication Users\". I need to check if I am (or another user is) in this group,
Accepted answer from DeanG is the preferred solution for getting this info within SQL Server
You can use Active Directory tools for this. I like Active Directory Users and Computers that is part of the Remote Server Administration Tools. Follow the link to download and install the tools on Windows 7.
Once installed, you can search for a specific group name:

Then you can see group membership using the Members tab:

If you don't want to use the AD browser packaged with RSA tools, there are several others available.