Spiders are a strange sort of mob- they have a number of characteristics that are completely different from the other mobs in the game, and drop not one but two very useful items.
Spiders come in two flavors- regular and cave. Cave Spiders are weaker and more frail than regular spiders, but inflict an annoying and dangerous venom on anyone they hit. They also do not spawn normally- they are only created by Spawners that can be found in abandoned mineshafts. Cave spiders can be identified by their blue color.
Either way, spiders function essentially the same.
Spiders are short, broad monsters, occupying a space two cubes wide and one cube tall. They can climb vertical surfaces- any kind of vertical surface, including leaves, gravel, glass, and fences just to name a few- just as though everything in the world were covered with ladders or vines. Spiders have no armor, and less health than Zombies and Skeletons (Cave Spiders have six hearts; regular Spiders have eight) and their attack is weaker than that of a Zombie or Skeleton (and Cave Spider attacks are weaker still).
However, because they jump when they attack, they deal extra damage- just as you do when you attack while falling or coming down from a jump. This and their ability to climb is what makes spiders especially dangerous. Spiders will often leap at you from just outside of melee range and, because of this, they not only become harder to hit, but they will not be knocked back as easily when you hit them- giving them more chances to hit you in return. Melee with more than one spider at a time is a dicey prospect at best, and a fatal one more often than not unless you’re especially well-equipped. Spiders can even spot you through solid blocks, making them fairly dangerous even when they’re in a parallel tunnel or on the surface overhead while you dig. They do tend to climb carelessly, though, and can be easily coaxed into falling to death.
There is an upside to dealing with Spiders, though- Spiders will not attack you if it is light enough. That is not to say that they will not fight- Spiders will always fight back if attacked, and if you encounter them in decent shade, they will still attack you. But if the day is clear and the light is strong, you can walk up to and even push around a Spider without any danger at all.
Spiders are a valuable opponent for other reasons, though. When you kill a Spider, it may drop a spider eye, a useful brewing component. Even more importantly, spiders tend to drop string when they die, which you can otherwise only get by using Clippers on cobweb blocks. String is needed for bows, fishing rods, other advanced tools- and, if you’re completely failing to find wool, you can use string to make wool blocks (though this isn’t advisable- wool is infinitely harvestable from sheep, but string is much harder to get).
While they are far more threatening than Zombies, and sometimes much more troubling than Skeletons, Spiders are far from the most dangerous thing out there- and they’re one of the most useful monsters to fight in the World, possibly only less important in that regard than Skeletons. Keep your watch keen for their red-glowing eyes and you’ll be well-served by your harvest of string.