The web strands come from what are known as spinnerets. These are four cone-shaped projections at the rear tip of the abdomen near the vent. The spider has complete control of the kind of webbing it exudes. It can be a single strand with no sticky glue on it, or it can be a swath used to wrap its prey, or it can be a strand with periodical globs of glue on it which its prey gets caught on. When the spider walks on its own web it only grabs onto the strands with no glue, otherwise it would get caught in its own web.
Spider web is the strongest filament, for its size, known to man.
