How does shutter speed affect a photo?

Shutter speed affects image quality in two principal ways. Firstly the longer the shutter is open (the slower the speed) the more light you get. The more light you get the less noise you have as it evens out the random emission of photons from a lightsource, and raises the signal produced by the sensor above the noise floor (the camera electronics produce a background level of noise, the bigger the signal you generate, by having more photons hit each pixel in the sensor, the less this background noise is noticed). This assumes you alter the other settings (aperture, ISO) to maintain the same exposure (overall brightness) in the image.

There is a point however when a long exposure can make an image noisier as it allows heat to build up on the sensor, which affects the recorded image.

Secondly shutter speed controls the impression of motion in an image. A long exposure/shutter speed blurs motion, giving the impression of greater motion of the camera or subject. It can enhance the image by making athletes or cars look like they're going very fast. It can detract from the image if the subject is not intended to be moving (e.g. a portrait) or if camera motion causes the image to come out blurred.