Just to illustrate the option that Lady Lunarius is referring to:
First, go to your profile settings by clicking "Profile" in the forum navigation bar.
Next, under the "Modify Profile" menu, click on "Personal Messaging", and you'll get to the screen below:
One of the options will read, Show a popup when I receive new messages.
If you check it and save your settings, you'll get a popup notification whenever you receive new PMs in your inbox.
Note that the popup won't come up if you aren't actively browsing through Elliquiy (e.g., you're typing a post for a roleplay, or only chatting in the Shoutbox but not browsing the forums). However, if you click "post" after you're done with your reply, or if after reading a long thread, you go to the next page, the popup will show before you can continue browsing.