Dreamweaver uses Javascript to display the popup window. If you take a look at the code, you can see the string "alert(msg)" which I assume sets this to the altert window you have mentioned. Therefore you could try to change the "alert" to something else... I don't write my own Javascript so I can't suggest anything but someone in the Javascript forum might be better able to help you with this one.
As for alternative suggestions, if I were viewing your web site, I would prefer such popups to appear in a separate window which I could then close afterwards. You can control more easily the properties of such windows in Dreamweaver and they are not as intrusive as alerts boxes etc. Of course this would be more work if you had a lot of definitions to do!
Other than that, dictionary.com is a good online resource. Try looking a word up and you will see in the address bar the standard link to words. You could then make a link from your web page to the relevant page of dictionary.com (check the Ts&Cs first to make sure they don't object to this - they prolly don't but there's bo harm in checking <g>) You could use the standard address and then just insert your own word at the end. You'll see what I mean when you try it.
HTH
Stuart