One way to do that is to copy the contents of the Target: edit field in the shortcut used to run Thunderbird and then add -profilemanager to it. Usually Thunderbird isn't on the path, so you will probably need to specify the program location when you do that. A window similar to the one shown will open. One way to do that is to type Thunderbird -profilemanager from the run line in Windows Start menu. (You can copy it anywhere, except the program directory.) Copy the profile folder to where you want to store it.
There is no OS X or Linux equivalent of the Windows start button for example but Linux and OS X users should be able to adapt the instructions as needed. Many of the examples/instructions in this KB article are written for Windows since most users use it. If you want to move a profile to a different PC, see Move to a new PC instead if you're using Windows.
See Dangerous Directories if that happens. This can cause a "The Local Directory path is not suitable for message storage" error message. For example, in C:\Windows or C:\Program Files. You can move it anywhere you want except the program directory, or a directory where Thunderbird thinks an application or the operating system might accidentally delete it. Normally you move a profile to store it in a more convenient location or to restore a backed up profile. It's recommended that you instead use a little known feature of the profile manager to move the profile. While it's possible to copy your profile somewhere and then edit that file to point to the profile, it's easy to make a mistake doing that. Thunderbird doesn't discover profiles by looking in the default location where profiles are stored instead it uses the profiles.ini file to find what profiles exist and where they are stored. This is the easiest way to find where your profile is stored. Help -> Troubleshooting Information -> Show Folder opens your profile directory using Windows Explorer (or its equivalent if you are not using Windows). Your profile has all of your mail, address books, settings, saved passwords, add-ons, etc. It stores it in a "profile" directory in order to make it easier to upgrade or re-install without worrying about losing your data. Thunderbird doesn't store your data in the program directory.