Seamless Pipe

Errr, that'd be a pipe with no seam down it.
If you take a long piece of flat plate, roll it until the edges meet and weld down the join you have a pipe, the long weld line is a 'seam'.
If you take hot metal and force it through a series of dies until it squeezes out as a tube it will have no seam or be 'seamless' as we sometimes call it.
Del

_{(I deserve a GA for avoiding any seems seams puns)}

A cheap and dirty answer would be, "Pipe that has no seams in it." However, it is actually rather tricky to describe fully, and it requires an understanding of manufacturing geometry.

First, you need to know that "skelp" is flat metal strip, which typically comes from a steel mill in rolled coils. This can be unrolled and formed into pipe in two ways: 1) It can be wrapped helically like a barber pole or candy cane, with each lap welded to the next to form a helical (spiral) seam. If you look around, you may see this in various places. 2) It can be wrapped longitudinally, like a hand-rolled cigarette, into a cylinder with one straight seam, which is usually done by electric resistance welding (ERW). This is an automated, high-quality process suitable for pressure piping. The seam is usually visible, but is fairly smooth and not very conspicuous.

Pipe or tube can also be extruded, which means that it is molten and is forced through a hole matching the cross-sectional shape of the finished part. If the part is solid (bar, rod, rectangle, angle, etc.), this is "easy." If the part is hollow (pipe or tube, whether round, square, rectangular, polygonal, elliptical, etc.), things get more complicated.

For hollow shapes, the outside of the shape is dictated simply by the hole it is pushed through. But the internal cavity of the hollow shape needs to go around a plug of some sort. This plug can be supported by metal bridges from within the chamber containing the molten metal, or it can be supported extrernally by means of a rod as long as the part to be extruded (in this case rod/plug are called a "billet" or "mandrel"). Here is where some visualization comes in.

In the first case (short plug supported by internal bridges or ribs), the molten metal passes over the bridges and then rejoins itself before passing through the extrusion outlet. The molten metal fusing back with itself constitutes a WELDED extrusion. There are seams that typically appear as striations on the finished piece. Although these welds are a potential weakness, the ribs and plug are rigidly attached to each other, and so the finished dimensions of the piece are pretty consistent.

In the second case, with the long external billet/mandrel, the molten metal passes through the extrusion outlet all as one continuous SEAMLESS mass. But now the extruded part must pass over the length of the billet, the extruded part must be cut off, the billet must be retracted from the part, and finally the billet must be repositioned for the next piece. The length of the billet means that it can wave around a bit, not necessarily being perfectly centered on the extrusion outlet. Because of this, the extruded part is seamless (and hence stronger), but the variation of wall thickness necessitates a tolerance factor (tolerance 12.5%, strength 87.5% of nominal size).

Welded pipes/tubes/extrusions are consistent in material thickness, but the weld strength may be reduced by a weld efficiency factor. Seamless parts have full material strength, but on account of thickness variations must be corrected with a tolerance factor.

I hope this gives some insight into the various manufacturing processes, as well as the reason for certain code provisions. Improvements to this exposition are welcome.

Although the kitty gave a suitable, short answer, ^{_{(though no GA for refraining from puns)}} your clear explanation deserves a GA.

Cheers.