A cobweb is the net of sticky threads that a spider weaves to catch insects. You can also call a cobweb a spider's web.
A cobweb is any spider's web, although some people use it to specify the dense, tangled type of web that certain spiders weave, or to mean a dusty old web, like the ones you might see in your grandparents' attic. Cobweb can also describe some intricate or complicated things: "She got lost in the cobweb of unfamiliar subway lines in the city." It comes from the Old English coppeweb, from the now-obsolete coppe, "spider."