A GIS integrates data with spatial information to make a visual representation, allowing for faster data analysis and more powerful data presentation. For a detailed, easy to understand introduction to GIS, see "What is GIS" from GISLounge.
Whatever area of the country (or world) you're interested in there are local entities (governmental, non-governmental and quasi-governmental) that collect and make data available. Search for local governments, community development organizations, regional planning commissions or government organizations, etc. Below are a few resources for data and statistics on the St. Louis Region.
Depending on your project, you may need to create your own data. However, there's a lot of data out on the web for free. Much of it is created and provided by government entities, some is provided by Universities or data clearinghouses that gather many sources of data in one place.
For information on Census data, see the U.S. Census Information Guide.
Here are some sources to get you started, but they are just the tip of the iceberg!
Even if you're not taking a GIS course on campus, ESRI has online courses to get you started and even teach you some of the more advanced features of ArcGIS. If you've used ArcGIS previously, and just want a refresher, or need to learn about a specific feature you haven't previously used, the short free training courses are a great place to start.
Anyone and Everyone! GIS is extremely flexible, allowing you to import existing data or create your own data. Any data with a geographic component can be used in GIS. See below for some examples of how GIS is used in various industries.