AHHHHHHHH OK
Yea. Dont worry. I did the same exact thing, as has pretty much everyone who uses myadmin.
It times out, or something, with big datasets.
Yep, from the shell, via putty in your case, run that command. Obviously replace out the relevant bits as it pertains to you.
To create the dump, its the exact same thing but > instead of <
Create the dump:
mysqldump -u root -p[root_password] [database_name] > dumpfilename.sql
Restore the dump
mysqldump -u root -p[root_password] [database_name] < dumpfilename.sql
The output will be created where you currently are, so I usually cd /tmp first, just in case I do something stupid.
dump it out, then transfer it to your new server if you are moving physically. Then restore it.
Frankly Mysqldump is slow and sucks for bigger nastier databases.
I use MK Parallel Dump
http://4sp.in/2lLand MK Parallel Restore
http://4sp.in/2lM