By default cron saves the output in the user's mailbox (root in this case) on the local system. But you can also configure crontab to forward all output to a real email address by starting your crontab with the following line:
MAILTO="yourname@yourdomain.com"
If you'd rather receive only one cronjob's output in your mail, make sure this package is installed:
aptitude install mailx
And change the cronjob like this:
*/10 * * * * /bin/execute/this/script.sh 2>&1 | mail -s "Cronjob ouput" yourname@yourdomain.com
Now that's easy:
*/10 * * * * /bin/execute/this/script.sh 2>&1 > /dev/null
Just pipe all the output to the null device, also known as the black hole. On Unix-like operating systems, /dev/null is a special file that discards all data written to it.