The health of your eyesight when using computer it really depends what kind of screen you're using and many other factors (DNA, diet, etc.), but in general, modern monitors (such as LCD, LED-backlit, white-LED, OLED) does nothing to the health of your eyes, unless you're still using CRT monitor. The bigger issue is rather neck and back if you don't site properly.
Remember, if you feel your eyes are not comfortable, you may adjust the brightness and contrast of your screen.
If you're using computer for too long, it's completely normal that your eyes would be tired and it really depends on the person (professionals can spent 8-12 everyday for years and their eyesight is perfect, for other 1h is enough).
In this case, you simply need a rest. It's usually advice to do short breaks and take a fresh air. You may also consider to train your eye muscles as a number of ophthalmologists believe that an exercise programme based on something called the Bates Method may keep eyes in better shape.
See: Can you really train your eyes to see better? where we can read:
Some of the principles of the Bates Method are already accepted by mainstream eye care.
More information about older monitors:
If you're using CRT most people experience mild discomfort unless the refresh rate is set to 72 Hz or higher. A rate of 100 Hz is comfortable at almost any size. It's usually advice to have a screen protector. However CRT are long lost technologies and this doesn't apply anymore to LCD monitors.