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 perform poorly, generate errors, crash, or the operation may succeed without an apparent problem but generate incorrect results.
Origins of bad geometry
The shapefile is an open format with a documented specification. Many software packages support writing to this format. Unfortunately, some of these software packages—sometimes due to bugs, sometimes due to lack of knowledge—do not follow the documented specification.
When shapefiles are loaded into a geodatabase, the features' geometries are copied as they are (with any existing geometry problems), so the same precaution that is needed when using shapefile data must be taken into account when using geodatabase feature classes. The exception to this is when the data is loaded to an enterprise geodatabase. Geoprocessing tools (including, but not limited to, Feature Class To Feature Class, Feature Class To Geodatabase, and Append) will check and repair the feature's geometry as part of the loading process.
DBMS systems provide native geometry field types. These types of fields can be uploaded and modified through manual SQL expressions or from other software packages. These systems can create geometries that are invalid for ArcGIS.
Responsibility
The onus is on the data's consumer to ensure that the feature class contains valid geometries before the data is used for 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. To fix these problems, use the Repair Geometry tool. See the particular tool's documentation for more details on what these tools do.