There are several types of data that are commonly available on the Internet for use in ArcGIS. A list of file formats supported by ArcGIS can be found here.
Shapefiles
Shapefiles are probably the most common format available for geographic data. They can be displayed by programs besides ArcGIS. A shapefile is actually several different files that must be kept together in order to function properly.
Geodatabases
The most common type of geodatabase available for download is a personal geodatabase. Geodatabases use Microsoft Access databases and can query and store spatial and non-spatial data. Shapefiles use dBASE tables to store attribute data.
Coverages
An older data format that was primarily used in ARC/INFO. Coverages have both spatial and attribute data. They often contain different classes of data, such as points, lines, and polygons.
It is important to note the geographic projection (or whether the data are unprojected) of any of the data you download from the Internet. This information is usually found in the metadata. ArcGIS has tools to reproject data, or it can do so "on the fly" in most cases. For spatial analysis, all data should have the same projection. More information on projections can be found here.
