i know some people rated dumb on this post, but Edward_great isn't wrong. But a few notes though-
You do not necessarily download the map. When you are in singleplayer, and you press ESC (by default unless you changed the key) to get to the game menu. You would then press "Save level...", in which it will prompt you to give the file a name, and then the game can export the map as .cw as someone said earlier in the post. If you are on web client, yes, the map will be downloaded and end up in your Downloads folder. However, if you are using the downloaded client, you will need to refer to your game folder to fine your saved map.
After that, you can use any cloud storage service to upload your file, but the most preferred method (and also for texture packs) is to upload your map file to Dropbox. After you've uploaded your file to the cloud storage service, make sure to obtain a URL to your file that would therefore be publicly accessible without being logged into the account. The URL should not exceed 64 or 128 characters (I forgot, someone correct me), in which most cases the site you are using will exceed the limit, which is why it is most recommended to use Dropbox.