The Spider Web is a location in the episode "Web Weirdos." Ed and Barb own the web, though Ed spun it. The Spider Web is the central location of the episode; the whole episode happens next to it aside from Ed's search for a gift and the very beginning of the episode.
The fact that Ed spun it turns out to be a subject of heated argument between the spider couple. Ed claims that Barb doesn't do her fair share of web spinning, and she gets mad at him for telling her to spin webs (she doesn't want to spin webs because she thinks Ed thinks she looks gross when she does).
Jake jumps into the web intentionally at first, calling it a "vertical trampoline." He then gets stuck prompting Finn to try to "stunt" him out, which consists of Finn jumping at it and also getting stuck. Later Bug and Fly are shown to also be stuck, waiting for the spiders to return and eat them. The web is so sticky it made even Jake's magic powers useless in escaping it.
The web was destroyed by Ed himself in the middle of an argument with Barb when he angrily decided to free the "food" to spite her. It is cut down using the Glowing sword. The only other time the web is cut is when Finn uses the beak of a bird he catches, though he only manages to cut a single strand.