Short Version
The number of textured pixels the card can render to the screen every second. To render a 3D scene, textures are mapped over the top of polygon meshes. This is called texture mapping and is accomplished by texture mapping units (TMUs) on the videocard. Texture fill rate is a measure of the speed with which a particular card can perform texture mapping.
How To Calculate It
Texture Fill Rate = (# of TMUs) x (Core Clock)
Does It Matter?
Though pixel shader processing is becoming more important, this number still holds some weight. Best example of this is the X1600 XT. This card has a 3 to 1 ratio of pixel shader processors/texture mapping units. And the X1600 XT gets creamed by the 7600 GT because of it. In the mid range, texture mapping can still very much be a bottleneck.
However, at the high end, the X1900 XTX has this same 3 to 1 ratio, but does just fine because screen resolutions top out and it has more than enough texture mapping power to handle any display.
Long Version
Over the years the ratio of texture mapping units to pixel pipelines has bounced around a bit. At first, the Voodoo1 had 1 pixel pipeline and 1 texture mapping unit. Then came the Voodoo2 which had 1 pixel pipeline, but 2 texture mapping units. This ability to apply 2 different textures in a single clock cycle was a huge advantage over other cards which would take 2 clock cycles to do the same job. But this was only an advantage in games that actually performed muti-texturing. (luckily several big games did and the Voodoo2 was a huge success).
The ratio stayed at 2 to 1 for quite a while, all the way up till ATi released the R300 powered 9700 pro which had 8 pixel pipes and 8 texture mapping units. The ratio changed here, because the number of pixel pipelines got so high that memory simply couldn't keep up. Even with the best memory of the day the 9700 pro's 8 texture units were underfed to begin with, if a second TMU per pipe had been added, it would have been completely starved and useless.
Recently ATi has introduced the X1600s and X1900s, both of which feature a 3 to 1 ratio of pixel pipes to texture units. This crippled the X1600, but gave the X1900s a nice boost (25% over the X1800 XT which was 1:1). We can see from this that a 3:1 ratio is somewhat inefficient (only a 25% boost in performance from a 300% boost in pixel pipelines). nVidia has remained at a 1:1 ratio ever since the launch of the GeForce 6 Series.