This article from Apple's support site gives specific steps for both OS X and Windows users on how to fix the problem.
In OS X, the problem can usually be fixed by simply selecting the folder, pressing Command+i (or right-clicking and selecting Get Info), then making sure that the permissions are set to Read & Write for all users and groups (you may need to click the lock icon and enter your password to make changes):
Also ensure that the changes you made will apply to all subfolders and files by selecting the option "Apply to enclosed items..." from the little gear menu:
If you are comfortable with Terminal, you can accomplish the same thing simply by entering the following command:
sudo chmod -R 1777 /Users/Shared