The '>' character is the prompt for a multi-line command. If you typed your command in exactly as you posted it in your message, you confused the shell. The backtick (`) and the single quote (') both have different meaning, and are not interchangeable. For what you were trying to do, you want to use two single quotes or two double quotes. You were also missing an option to find. You should try this:
find -iname "*mysql*"
That command will look in the current directory and all subdirectories for any file/directory that contains "mysql" in the name (in any combination of lower or upper case characters). You could also do:
find / -iname "*mysql*"
That command does the same thing except
the starting directory is specified (the root directory in this case: / )
Use this command to read about what find can do (some of it will probably be a little overwhelming if you aren't familiar with man pages):
Now, if you're really interested, here's the reason the shell got confused. When you used an opening single quote character without a matching closing one, the shell assumes you plan to enter more text and eventually provide the closing single quote. So when you hit enter, it gave you another line to finish the command. That would continue until you supplied the expected closing single quote.
The backtick character has special meaning that isn't necessary to go into right now
It's probably not what you need at the moment.