Winning a grant requires writing a well-reasoned and respectful argument for funding one's project. A grant is a gift, usually from the taxpayers, and we are expected to explain why a project makes sense and has the potential to yield useful results. Success also requires a bit of luck, given the inevitable vagaries of budgets and the review process.

That said, here are some pitfalls to avoid, and some specific suggestions for improving your chance of success.

1. Choose an important problem. However, one needn't promise to solve the whole thing. It is often sufficient to aim for significant progress on a part of the larger challenge.
2. If you have questions, ask the granting agency. Granting agencies want to receive high quality proposals that will advance their goals. It is almost always appropriate to contact their staff for information regarding what types of projects they wish to fund and on how you may be able to frame a project that they would seriously consider. Agency staff can also help you understand how the review of your proposal might be handled and may give you the opportunity to make suggestions, such as the most appropriate review panel for your proposal. They can also help you understand many other aspects of the application and review process, such as what level of detail to include in a budget justification and whether there are any common gotchas to avoid. On the other hand, don't contact them so often that you become an annoyance. And never try, or even appear to try, to inappropriately influence the review process.
3. Include a crisp list of goals. Write your chief goals (e.g., NIH specific aims) as a short list so that reviewers can quickly grasp the basic thrust of the proposal. (Admittedly, there is usually a level of arbitrariness in how the proposed work is broken into a discrete list of items.) Three to six items are probably typical for an NIH R01.
4. Include some goals that are like to be achievable and place them at the top of the list. These may not be very exciting, but they guarantee that at least part of your project will be work.
5. Include some goals that are more thought-provoking and innovative, although they may be less likely to succeed, and put them at the end of the list. These may or may not work out, but they will make your proposal more exciting and create the potential for a big payoff.
6. Display knowledge of the field. Make clear in the Background section or its equivalent that you have a good grasp of the field and the state of the art, and make sure that what you propose neither repeats nor contradicts previous work, unless you provide a good reason for doing so.
7. Do not gloss over or ignore potential weaknesses of your proposed approach. The reviewers will probably find them anyhow and may judge you as incompetent or even dishonest for not acknowledgeing and addressing them. It is much better to explicitly address potential concerns. Possible approaches include arguing that your work will advance the state of the art despite the obstacles, that you have a solution at least worth trying, or that the apparently problem does not actually matter for some other reason.
8. Don't propose anything that violates or claims to disprove tenets of the field without providing a strong rationale for doing so.
9. Make sure your budget is connected to reality. Even if the instructions do not call for a detailed justification (e.g., an NIH modular budget), the reviewers have a sense for the costs of personnel and supplies and they will probably estimate your budget needs on their own and compare with your request. Don't ask for more than you need, but don't underbid either: you can't do the work without adequate funding.

   If your proposal is funded, but at a significantly lower level than you reasonably requested, you may have trouble completing the proposed work. In such cases, the granting agency may allow you to delete one or more Aims or goals from the actual work. This has the advantage of providing adequate funding for the reduced scope of work. It also frees you to include the deleted Aims in a separate proposal.

10. Create a high-quality document. Write well. Proofread and have a friend proofread and critique the text, unless you are confident in your writing skills. Make sure figures are legible even in black-and-white and after photocopying, since some reviewers probably will be working from copies or from printouts on their own black-and-white printers. A poorly written or sloppy proposal creates a negative impression that can be hard to overcome. Reasonably or not, it raises the question whether the research also will be done sloppily.