A pipeline is like an assembly line. Ex. A car assembly line. The frame of a car moves down the line and people keep adding the pieces until it eventually becomes a car.
A pixel pipeline is a pipeline that outputs one pixel. A pixel pipeline working at a frequency such as 1MHz can output one million pixels per second. GPU designers would design a GPU with more than one pixel pipeline because it was more feasible than making just one pixel pipeline at an insane frequency.
Pixel shaders are functions that can modify a pixel. Pixel shaders are used extensively in today's games: in water, bump mapping, and other graphical effects.
Previously, when pixel shaders were starting to become more common in games, the definition of performance in a GPU changed. It went from Texels/sec to Pixels/sec. The reason is that for every clock cycle of a pixel pipeline a pixel shader operation can be accomplished. Although, I'm not saying that most shader operations are processed in one cycle.
This worked fine until designers started to realize that when they perform a pixel shader operation, other important functions of the pixel pipeline aren't being used. They viewed this as wasting power, hence the invention of Pixel Shader Execution Units. I don't know if that's the real name, but basically all they do is calculate pixel shader operations, freeing up the pixel pipelines.
Since pixel shader performance is the main GPU bottleneck in good framerates in today's games, hardware Pixel shader units are now the definition of performance of a GPU.