Left to right? — If people read in languages that move from left to right, then they tend to look at the screen from left to right. If they read from right to left, it is the opposite.
Not the edges — People tend to ignore the edges of screens. Because people have gotten used to the idea that there are things on computer screens that are not as relevant to the task at hand, such as logos, blank space, and navigation bars, they tend to move towards the center of the screen and avoid the edges. After the first look at a screen people then move in whatever is their normal reading pattern, in other words left to right/top to bottom in cultures that read that way.
Grabbing attention — If there is something that grabs attention, for example, a large photo (especially one with someone’s face), or movement (animated banner, video) somewhere else on the screen, then you can pull them away from their normal reading path and get them to look elsewhere, at least briefly.
Where to find certain tools and features — People have also gotten used to the location of certain items on a screen. For example, navigation bars are usually on the left or the top. Logos are at the top left. Search is expected at the top, either in the middle or towards the right. Help links or buttons are usually at the top right.
What do you think? Is it important to design with these conventions in mind? Or do you sometimes break out of the mold?