4. Mathematics
Having a good understanding of mathematics is crucial to success in economics. Most undergraduate students, particularly those coming from North America, are often shocked by how mathematical graduate programs in economics are. The math goes beyond basic algebra and calculus, as it tends to be more proofs, such as "Let (x_n) be a Cauchy sequence. Show that if (X_n) has a convergent subsequence then the sequence is itself convergent". I've found that the most successful students in the first year of a Ph.D.
program tend to be ones with mathematics backgrounds, not economics ones. That being said, there's no reason why someone with an economics background can not succeed.
Mathematical Economics Material You Must Know as a Bare Minimum
You'll certainly want to read a good undergraduate "Mathematics for Economists" type book. The best one that I've seen happens to be called Mathematics for Economists written by Carl P. Simon and Lawrence Blume. It has a quite diverse set of topics, all of which are useful tools for economic analysis.
If you're rusty on basic calculus, make sure you pick up a 1st year undergraduate calculus book. There are hundreds and hundreds of different ones available, so I'd suggest looking for one in a second hand shop. You may also want to review a good higher level calculus book such as Multivariable Calculus by James Stewart.
You should have at least a basic knowledge of differential equations, but you do not have to be an expert in them by any means. Reviewing the first few chapters of a book such as Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima would be quite useful. You do not need to have any knowledge of partial differential equations before entering graduate school, as they are generally only used in very specialized models.
If you're uncomfortable with proofs, you may want to pick up The Art and Craft of Problem Solving by Paul Zeitz. The material in the book has almost nothing to do with economics, but it will help you greatly when working on proofs. As an added bonus a lot of the problems in the book are surprisingly fun.
The more knowledge you have of pure mathematics subjects such as Real Analysis and Topology, the better. I would recommend working on as much of Introduction to Analysis by Maxwell Rosenlicht as you possibly can. The book costs less than $10 US but it is worth it's weight in gold. There are other analysis books that are slightly better, but you cannot beat the price. You may also want to look at the Schaum's Outlines - Topology and Schaum's Outlines - Real Analysis. They're also quite inexpensive and have hundreds of useful problems. Complex analysis, while quite an interesting subject, will be of little use to a graduate student in economics, so you need not worry about it.
Advanced Mathematical Economics that would be Helpful to Know
The more real analysis you know, the better you will do. You may want to see one of the more canonical texts such as The Elements of Real Analysis by Robert G. Bartle. You may also want to look at the book I recommend in the next paragraph.
What Advanced Mathematical Economics Book You'll Use When You Get There
At the University of Rochester we used a book called A First Course in Optimization Theory by Rangarajan K. Sundaram, though I don't know how widely this is used. If you have a good understanding of real analysis, you will have no trouble with this book, and you'll do quite well in the obligatory Mathematical Economics course they have in most Ph.D. programs.
You do not need to study up on more esoteric topics such as Game Theory or International Trade before you enter a Ph.D. program, although it never hurts to do so. You are not usually require to have a background in those subject areas when you take a Ph.D. course in them. I will recommend a couple of books I greatly enjoy, as they may convince you to study these subjects. If you're at all interested in Public Choice Theory or Virginia style Political Economy, first you should read my article "The Logic of Collective Action". After doing so, you may want to read the book Public Choice II by Dennis C. Mueller. It is very academic in nature, but it is probably the book that has influenced me most as aneconomist. If the movie A Beautiful Mind didn't make you frightened of the work of John Nash you may be interested in A Course in Game Theory by Martin Osborne and Ariel Rubinstein. It is an absolutely fabulous resource and, unlike most books in economics, it's well written.
If I haven't scared you off completely from studying economics, there's one last thing you'll want to look into. Most schools require you to take one or two tests as part of your application requirements. Here's a few resources on those tests:
Get familiar with the GRE General and GRE Economics Tests
The Graduate Record Examination or GRE General test is one of the application requirements at most North American schools. The GRE General test covers three areas: Verbal, Analytical, and Math. I've created a page called "Test aids for the GRE and GRE Economics" that has quite a few useful links on the GRE General Test. The Graduate School Guide also has some useful links on the GRE. I would suggest buying one of the books on taking the GRE. I can't really recommend any one of them as they all seem equally good.
It is absolutely vital that you score at least 750 (out of 800) on the math section of the GRE in order to get into a quality Ph.D. program. The analytical section is important as well, but the verbal not as much. A great GRE score will also help you get into schools if you have only a modest academic record.
There are a lot fewer online resources for the GRE Economics test. There are a couple of books that have practice questions that you may want to look at. I thought the book The Best Test Preparation for the GRE Economics was quite useful, but it's gotten absolutely horrid reviews. You may want to see if you can borrow it before committing to buying it. There is also a book called Practicing to Take the GRE Economics Test but I've never used it so I'm not sure how good it is. It is important to study for the test, as it may cover some material that you did not study as an undergraduate. The test is very heavily Keynesian, so if you did your undergraduate work at a school heavily influenced by the University of Chicago such as the University of Western Ontario, there will be quite a bit of "new" macroeconomics you'll need to learn.
Conclusion
Economics can be a great field in which to do your Ph.D., but you need to be properly prepared before you enter into a graduate program. I haven't even discussed all the great books available in subjects such as Public Finance and Industrial Organization. I'm sure I've also left out more great resources than I've included.