Geodatabase: The default data structure for recent versions of ArcGIS, intended to better manage individual layers, tables, annotations, and other files that are part of the relational database. The geodatabase will show up on your computer as a folder with the .gdb extension.
Shapefile: The native ArcView 3.x format, but still commonly used, these have the extension .shp. When you open one of these layers, it also uses several auxiliary files that are in the same folder as the .shp file (i.e., several related files are needed for the shapefile to open). Probably the most common ArcGIS files freely available.
Layer File (.lyrx extension) and Layer Package (.lpkx): The Layer File saves layer properties outside of a map or the project environment. The Layer Package saves the properties as well as the underlying data, for sharing with others who don't have access to the referenced data.
Raster Image Formats: ArcGIS Pro can open selected raster dataset formats. If not already spatially aware, the image (e.g., a scanned map) can be georeferenced, such as certain JPEG files or the GeoTIFF format. See documentation on georeferencing.
ArcGIS Pro Project (.aprx extension): This is the ArcGIS Pro working environment for building maps and layouts, and for doing analysis. Can be shared when packaged as a Project Package (.ppkx). There are other sorts of packages in ArcGIS Pro that are intended for sharing maps and/or data.
Map File (.mapx extension) and Map Package (.mpkx): A separate ArcGIS Pro map, outside of the Project environment, intended for sharing. The Map File does not include underlying data The Map Package includes a Map File as well as it's underlying data, for sharing with ohters.
Map Document (.mxd extension): These must be imported into an ArcGIS Pro format to use in ArcGIS Pro. This is a file representing a map workspace created in ArcMap Desktop (NOT ArcGIS Pro). Can also be packaged with its data, for portability, as a Map Package (.mpk), which also can be imported into ArcGIS Pro. Detailed tutorial on importing these formats. It's sort of a bookmark file, which tells the program which layers and tables are being used for the particular map and how they are being symbolized. All the files referred to that are being used to build the layers and store the data must be accessible.
Smart Data Compression (.sdc extension): ArcGIS pro does not support this format; you must convert to another format using ArcMap Desktop. This is used, for instance, for layers distributed in the old ESRI Data & Maps product. Data must be exported from within ArcMap to a feature class (shapefile or geodatabase layer) to use with analysis tools in ArcMap or to use with ArcGIS Pro. Conversion Tools in ArcMap Desktop
Coverage: ArcGIS pro does not support this old Arc/INFO format. You must use ArcMap Desktop. Each coverage is stored in its own unique folder. The folder's name is the name of the coverage; the files within each folder have standard names (arc, arx, pat, dbx, etc.).
ARC/Info Interchange (Export) Format: ArcGIS Pro does not support this format, which was used to package Arc/INFO coverages for sharing. This format, whose extension is usually .e00, will need to be translated into ARC/Info coverages in ArcMap Desktop. You can use the Import from Interchange File tool