From Nookipedia, the Animal Crossing wiki
The spider (クモ, Kumo) can be found dangling from fruit-bearing trees once it has been shaken by the player. They are common and easy to catch, since they do not try to escape from the player if they are approached.
Tarantulas are another kind of catchable spider in Animal Crossing games, but in-game, they are not related to spiders.
 Donating to the Museum
 In Wild World
"I can think of nothing more repugnant than becoming entangled in a spider's web! After thrashing about, you think you got it all off, but no, indeed! A little tickle on your neck and you thrash anew! Blech! Such foulness... Hoo... But I digress..."
The spider is displayed in the second room of the insect exhibit, on the second tree from the bottom at the far end. It is seen descending from its hiding place in the leaves of the tree on a spider silk, waiting a while, and then scrabbling back up again.
 In City Folk
"Horrors! My apologies, but I get rather jumpy around fast, multi-legged critters like this one, eh wot? Even worse is the fact that spiders have eight eyes! Even when it looks away, it can see our every move! I've heard some people worship the spider... And I've also heard that you will not find me in those places."
After donating, the spider can be found on a tree in the central terrace of the insect exhibit. It will come down out of the tree on a thread when approached.
 Encyclopedia Information
When the player has caught the spider, they can find information about the insect in the bug menu. Below is the information regarding the spider:
 Wild World
 City Folk
 Further Information
Argiope bruennichi, commonly known as the wasp spider, is a widespread species of spider that can be found in Europe, Africa and much of Asia. Their common name refers to the black and yellow stripes they have on their abdomens,
 Names in other languages