I didn't find The Universe in a Nutshell by Stephen Hawking enlightening at all, so I doubt A Brief History of Time is any better. I didn't even finish the book because of its lack of real content. So I'm not impressed by Hawking's writing, as it is very superficial. His books are more for the general public than someone actually interested in physics. You would be much better off reading everything by George Gamow and Richard Feynman. They are much more interesting and fun to read than Hawking, plus their texts contain oodles of substance. I would start off with Gamow.
I also recommend Understanding Physics by Isaac Asimov. This book requires no calculus and only high school math, but it contains a lot of interesting physics. It is very thick and comprehensive, so it will keep you busy. Gamow's books are fun because they are shorter. Plus, he writes so well that they are a quick read.
Just to give some more background I do know calculus (through multi variable) so I can handle a book that has some rigor to it. That being said, I do not want a book that is too dry or something that is more of a pain than enjoyment. Ill look into gamow and feynman.
Then you might give Feynman's lecture series a try if you're interested in physics. I've never read them, but it might be tough going at first if you don't know much physics. I really don't know though.
Either that or you can try out a vector calculus book to learn some very important math. I would recommend http://matrixeditions.com/UnifiedApproach4th.html" by John Hubbard. If you can make it through this book, you will be light years ahead of your fellow physics, math, or engineering majors if that's what you end up doing.
How solid of a background? Intro physics? More? I had always thought it was a book in laymen's terms...
Most likely high school physics would be a great help. Though with it was made to be readable in laymen's terms, its not that simple to understand. Especially if your looking to go into advanced topics. His arguments for things always make my head spin. Though you should also know some cosmology. The book is centered around a lot of their concepts.
You'll be able to read every single popular account of physics without any mathematical background & understand most of them.
I've read so many before & understood most of them well, a few did get crazy for a while but it's bearable. Keep in mind this at a time when I had forgotten even how to add fractions :p so it's possible without a background in phys & math.
Btw, get a BriefER History of Time. It's just an updated version.