It helps to have basic grammar and prose skills, but that's only part of it.
As for resources, pick up "The Elements of Style" by Strunk & White. That will give you a good insight into writing only what is necessary. Try studying some of David Ogilvy's work, or even pick up a copy of "Confessions of an Advertising Man." I have read a few kitschy books like "Cashvertising!" which actually proved to be quite helpful in describing the ins and outs of effective ad copy.
Just remember that the main purpose of ad copy is communication. You need to get your point across without misleading or confusing your audience.
You can also improve your copy by improving the way you go about writing it. I personally believe anyone can become a great writer. Identify what you want to communicate and who you want to communicate it to. For the same reason that design is only effective if it truly serves the user, copy is only effective if it truly speaks to the reader.
It might seem obvious, but keep your basic message and your audience in mind at all times and the words will come easily.