There is no simple way to "maximize" your windows to fill the whole screen in OS X...that's not the way things are supposed to work. You can drag the lower right corner of the window to mostly fill the screen though.
Windows and OS X have different philosophies when it comes to multi-tasking. Windows is more-or-less designed to run one app at a time, so it likes to fill your entire screen. You then have to use the taskbar/alt-tab to get to a different app. OS X prefers that you have depth to your workspace, and so wants to leave space for you to be able to click on other apps and windows behind the active one, which is why it doesn't like to fill your entire screen.
Many people mistakenly think that the green button in OS X windows is a "Maximize" button...it's not. Apple calls it a "Zoom" button, and it automatically sizes your window to optimally (in the opinion of the programmers) fit the window's content.
In this day of ever larger monitors with ever higher resolution, this is now the most sensible way to do things...a browser window shouldn't have to fill your entire screen in order for you to see everything. You can now easily have a browser window, an iChat buddy list window and other things all visible at the same time.
