A thorough discussion of various formats that ArcGIS can read may be found in the help documentation.
Detailed information on formats for geospatial data, many of which can be used in ArcGIS, has been compiled as part of a Library of Congress initiative on planning for the sustainability of digital formats. The website also includes a discussion of other aspects of geospatial data.
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. If all the layers being used for a project are inside the geodatabase, then the geodatabase folder and all contents should make the entire project easy to move and share. More information on geodatabases.
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, and easy to work with.
Coverage: ArcGIS will also read coverages in ARC/Info format. 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.).
Layer File (.lyr extension): Not a format that stores data, but used within ArcGIS 8 and higher to store references (e.g., full path) to actual data (e.g., Shapefiles) and also to store how the data is displayed, associations, etc.
ARC/Info Interchange (Export) Format: This format, whose extension is usually .e00, will need to be translated into ARC/Info coverages. You can use the Import from Interchange File tool.
Smart Data Compression (.sdc extension): An efficient read-only format for storing a layer, but can be directly accessed within ArcGIS. Used, for instance, for layers distributed in the ESRI Data & Maps product. Data must be exported to a feature class (shapefile or geodatabase layer) to use with analysis tools.
Raster Image Formats: ArcGIS can open selected raster dataset formats. These may be georeferenced (registered), such as certain JPEG files or the GeoTIFF format. Even if an image is not registered, they can be manually georeferenced so you can overlay other layers. See documentation on the official help pages.
Map Document (.mxd extension): A file representing a map workspace created in ArcMap. 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.