When you setup email accounts using the "IMAP" protocol, everything is synchronized: when you move an email to a folder using Windows Live Mail, the change is reflected on the server; and if you delete a message, it will be deleted on the server as well. The disadvantage of this approach is that your mail server basically keeps a copy of all messages, even if they're no longer needed "live". The Leave a copy of messages on server option is useful for this scenario for email accounts you've added as "POP3", which allows you to keep emails on the server and on your computer completely detached.
FYI: AOL Mail, Hotmail / Outlook.com and Gmail are added to Windows Live Mail as IMAP (setting doesn't apply). Yahoo Mail is added as POP3. You can tell by the account's properties (explained below) - if no Advanced tab is visible, or if that option is missing from it, you're not connecting with POP.
Each email account can have its own setting when it comes to leaving copies of emails on the mail server: to configure them, the easiest way is to click on the File button (top left corner) - then, choose "Options", and "Email accounts" on the right: 
Just double-click on any account listed, and its Properties window will open.
Tip: you can also right-click on an account on the left (folder listing) and choose "Properties". (Right-click on the account header, not one of its folders!)
In the account Properties window, select the Advanced tab: it you don't see it, it means that the account uses IMAP (an IMAP tab will be visible), or another IMAP-like protocol like Microsoft's (Exchange) DeltaSync. Here are the properties tabs for two such accounts - Gmail on the left, and Hotmail / Outlook.com on the right: 
For all other POP3 accounts, click on the Advanced tab: if the "Leave a copy of messages on the server" checkbox is checked (it is by default in Windows Live Mail 2012), a duplicate copy of all downloaded emails will also remain on the mail server.
But notice the default option configuration: 
• By default, the "Remove from server when deleted from 'Deleted Items'" checkbox is checked. When you delete messages in Windows Live Mail, they go to a "trash" folder (just like deleted files on Windows), called "Deleted Items
", and each email account has its own Deleted Items folder. When that checkbox checked, your deleted messages will be removed from the server as soon as you empty the Deleted Items folder, by deleting individual emails or using the "Empty Deleted items folder" command: 
• The "Remove from server after [x] days" checkbox is unchecked by default: unless your mail server is seriously running out of space, or you have to follow a specific company policy, we strongly suggest that you leave setting disabled. Sooner or later, you will regret having had your email messages (important or not) automatically removed from the server after an arbitrary number of days.
If you do enable this option, you'll be able to set a custom number of days: 
Once you have configured these options to your liking, click "OK" to save them, and optionally repeat for each POP3 email account you have setup in Windows Live Mail.
