The Export menu, found on the main menu, contains a few tools that allow you to extract layers from your map and make them available to other programs.  Click any of the options below for more details.

Export to Shapefile
Export a layer to an ESRI Shapefile.

Export to CSV file
Export a layer to a spreadsheet file.

Export Attachments
Export all pictures and documents attached to a layer

Export to DML file
If you are sending a copy of a layer to another user of this same GIS software then use this export option.  The DML file format will not, however, be useful to people using other GIS software.

See Geodatabase Tools for instructions on exporting to an ESRI Geodatabase

See Import menu also.