A feature class is a single geometry with an attribute table...period. True, it should also have a spatial reference, and likely needs one to function correctly. Still, in the simplest form, all you need is a single geometry (points, lines or polygons) and an associated attribute table.
It is no great secret I am an instructor for Esri. As a result, my perspective of GIS is highly Arc-Centric. It could even be said I view GIS using the Esri projection, but that might be a little too geeky even for me. I digress. Let's get back on topic. Focus monkey...focus.
As an instructor, I teach a huge variety of classes covering just about everything Arc-related. Almost universally, whether the class is an introduction or an advanced class, there is at least one person in the class who possesses a bit of a misconception about exactly what a feature class is. A feature class is a single geometry with attributes.
A single geometry refers to either points, lines or polygons. Each feature class stores only one of these three geometries. True, if you want to split hairs, there are others. However, these three are far and above the most common.
The associated attribute table stores all the other information you need to know about the features depicted by the geometry, the points, lines or polygons on the map. If talking about land records, land parcels, we might want to know the assessed value, current owner, address, purchase date and purchase value.
Why does this matter? Until you start geoprocessing, it likely does not. When digging into ArcToolbox though, there are quite a few tools requiring feature classes as inputs. Specifically, when importing vector data into a geodatabase, the Feature Class to Feature Class tools are used. There is one tool for importing a single feature class and another for batch importing. These tools can take any feature class and import it into the geodatabase. What then, can these tools use as input? They can ingest any feature class.
This brings us back to the definition of a feature class, a single geometry with an associated attribute table. Based on this definition, is a shapefile a feature class? What about single geometry in a dwg file (CAD)?
The answer in both cases is yes. They are both feature classes. A Shapefile is a standalone feature class. A single geometry in a dwg file is simply a feature class in a CAD file. Both fit the definition. They both possess a single geometry with an associated attribute table. As a result, both can be used with the Feature Class to Feature Class tools.
There you have it. A feature class is comprised of a single geometry and an associated attribute table. It can be a feature class in a geodatabase, a shapefile, or a feature class in a CAD file. All of these can be used as inputs into the Feature Class to Feature Class tools when importing vector data into the geodatabase.