ArcGIS applications are built with the assumption that the feature's geometry follows certain specifications. When the processing algorithms encounter data that does not follow the specifications, the software can return errors or crash, or the operation may succeed without apparent problem but the result may be incorrect.
Origins of bad geometry
The shapefile is an open format to which many software packages write. Unfortunately, some of these software packages—sometimes due to bugs, sometimes due to lack of knowledge—do not follow the documented specification of the shapefile format.
Also of note is that when shapefiles are loaded into personal or file geodatabases, the features are copied as they are (with any existing geometry problems) to these other formats, so the same caution and care that need to be used when using shapefile data must be taken when using personal and file geodatabase feature classes.
The exception to this is when the data is loaded to an ArcSDE geodatabase. Before it is stored in the feature class, each record's geometry is checked for geometry problems and repaired if necessary.
The onus is on the data's consumer to ensure that the feature class contains valid geometries before the data is used in projects or analysis. Once Check Geometry has been run on a given dataset, it is unnecessary to check it again until the data is modified.
Finding and fixing geometry problems
The Check Geometry tool will generate a report of all features with geometry problems within the feature classes provided. To fix the problems, use the Repair Geometry tool. See the documentation for these tools to learn which geometry problems are checked for and addressed.