I have a shapefile that I would like to save to a specific catalog database. I created a new database connection in arc catalog but I am not sure how to save this shapefile that was sent to me via email to my new catalog. Ultimately I want to export the database table with that shapefile data(attributes) to a separate SQL file.
If your destination is an ESRI geodatabase (enterprise geodatabase or file geodatabase).
Use the "Feature Class to Feature Class" tool, or the "Copy Features" tool.
For more information (including the difference between the two tools) see: Difference between Copy Features and Feature Class to Feature Class tools?
OR, if you only want the attributes, and not the spatial information, use the "Table to Table" tool.
If your destination is NOT an ESRI geodatabase, load up the shapefile in ArcMap, and right click on the layer in the table of contents, and then choose "Open Attribute Table". Then in the attribute table's menu (top-left icon), choose "Export".
You can export to text/CSV or to DBase format, either of which should be able to be imported into an SQL database using your favourite SQL tools.