How do I crop a section of the US map I am working on from the larger DEM data and save as a separate file on ArcMap 10.2?
Create a Polygon Boundary of your area of study then use the polygon as your clipping boundary in the Raster Processing Clip Tool.
In ArcToolbox: Select
Data Management Tools>Raster>Raster Processing>Clip
GIS Geography provides a good article on How to Clip Rasters to Polygon Boundaries in ArcGIS
I find that simple is best... Using the drawing tool draw a graphic box around the area you want to clip. Right mouse on Image and select "Data" then "Export Data" Pick you image type and select OK.
You want to use the Clip tool for this. There are several ways, the least RAM intensive way for a large file:
In the image analysis window, use the clip area tool for a general space around the region of interest you have. This will create a temporary layer file.
Create a new polygon for the study area you're looking at, which will be a polygon shapefile. Do this within the arc catalog window within Arcmap [right click on an empty folder/geodatabase; select 'create shapefile]
Open the edit feature toolbar, and select your newly created shapefile to be edited, draw a polygon and save the edit: this is your new study area
run the 'clip' tool, with the new study area as the polygon to clip your temporary image layer to ---> this will be your final DEM image clipped to the area you need.