A telescope's spatial (or angular) resolution refers to how well it can distinguish between two objects in space which are separated by a small angular distance - it is the telescopes ability to resolve fine detail. The closer two objects can be while still seen as two separate objects, the higher the spatial resolution of the telescope. The spatial resolution of a telescope affects how well details can be seen in an image. A telescope with higher spatial resolution creates clearer and more detailed images. As an analogy, lets think of our eyes as being visible light detectors. A person's eyes have high spatial resolution if, for example, they can see the letters on a page as being sharp, clear and distinct.
A lower (left) and higher (right) resolution image of the Andromeda galaxy
The spatial resolution of a telescope depends on the size of its lenses or mirrors and the size of the pixels in its detectors. The resolution is also limited by air turbulence (for ground based observatories) and by the smoothness of a telescope's mirrors or lenses. The spatial resolution of a telescope is proportional to the wavelength of light being detected divided by the diameter of the telescope. Larger telescopes have better spatial resolution. However, it is the size of the telescope relative to the wavelength that really counts. The longer the wavelength, the larger the telescope needs to be to get good resolution.
Spatial resolution is usually measured in units of arcseconds or arcminutes. Our eyes, for example have a spatial resolution of about one arcminute.
IRAS (left), ISO (center) and Spitzer (right) simulated view of the same region of space