How can we see stars if their apparent width is less than a pixel? Stars are so far away that their apparent width is essentially zero when compared to any pixel of a camera or TV screen.
And yet we can still see them.
According to our eyes stars have a finite albeit very small width. Why is this, and what is this width?
 A: 
Stars are so far away that their apparent width is essentially zero when compared to any pixel of a camera or TV screen.
And yet we can still see them.

In an electronic sensor (CMOS or CCD for example), the pixel sums all the light that reaches it. Even if light only falls on a small part of the actual sensitive area of the pixel, it will generate photocurrent that the camera will record as part of the image.
Of course if the image of the star on the sensor is small enough, it might fall in a gap between the pixels. But there are also resolution limits to the optics forming the image that generally prevent this being an issue.

According to our eyes stars have a finite albeit very small width. Why is this, and what is this width?

This is the resolution limit of our eyes, determined mainly by diffraction from the aperture of the pupil of the eye.
A: The light beam from a very distant star wiggles about in position by a tiny amount due to the effects of passing through the earth's uneven atmosphere on its way towards your eye. In this way it then excites more than just a single photoreceptor in your eye or pixel in an imaging device and thereby produces a "visible" signal. 
Note also that a telescope concentrates the light from a small patch of the night sky to the point where the light from faint objects in that patch becomes strong enough to trigger photoreceptors in your eye or pixels in an imaging device. The more powerful the telescope, the greater this magnification effect is, and the easier it becomes to capture enough light from a faint source for you to then see it with your eye, or make an electronic image out of it- which is exactly what the Hubble space telescope does. 
