Your local machine is the computer upon which you are working. Saving data to your local machine is different than saving it to an online server, a USB flash drive, or any other external device. You should back up the data on your local machine often.
When downloading data from the internet or from e-mail you are taking data off of the server where those files are stored and copying them to your own local machine. This takes up space on your hard drive and can be accessed when your computer is online or offline.