Texture Mapping Units (TMUs)
Textures need to be addressed and filtered. This job is done by TMUs that work in conjunction with pixel and vertex shader units. It is the TMU's job to apply texture operations to pixels. The number of texture units in a graphics processor is used when comparing two different cards for texturing performance. It is reasonable to assume that the card with more TMUs will be faster at processing texture information.
Raster Operator Units (a.k.a. ROPs)
The raster operation processors are responsible for writing pixel data to memory. The speed at which this is done is known as the fill rate. ROPs and fill rates used to be a much more important metric in the early days of 3D graphics cards. While the job of the ROPs is important, it is not really a performance bottleneck as much as it once was, and is not used as a relative performance indicator to good effect at this time.
Pipeline is a term used to describe the graphics card's architecture, and it provides a generally accurate idea of the computing power of a graphics processor.
A pipeline isn't formally accepted as a technical term. There are different pipelines within a graphics processor as there are separate functions being preformed at any given time. Historically, it has been referred to as a pixel processor that is attached to a dedicated TMU. Graphics cards like the Radeon 9700 had eight pixel processors, each attached to a single TMU, and as such it was considered an eight-pipeline card.
The term pipeline no longer accurately describes some of the newer graphics processor architectures. Processors have a fragmented structure compared to past designs. ATI, with its X1000 series graphics card, was the first to deviate from this norm in order to achieve performance boosts through substructures optimizations. Some units are used more than others, and in an effort to increase the processor's entire performance, they attempted to find a "sweet spot" in the number of units needed for optimum efficiency without the need for excess silicon. In this architecture the name pixel pipeline lost its meaning as pixel processors were no longer attached to single TMUs. For example, ATI's Radeon X1600 graphics processor has 12 pixel shader units and only four TMUs. It can not accurately be described as either a 12-pipeline architecture, nor can it accurately be described as a 4-pipeline architecture, although it is often referred to as either.
As such, the number of pipelines in a graphics processor is usually used to compare two different cards (other than ATI's X1x00 series). For example, when comparing a card with 24 pipelines and a card with 16 pipelines, it is reasonable to assume that the graphics card with 24 pipelines will be generally faster.