The Web of Destiny is a collection of challenges that must be completed within each level. There are 15 per level for a total of 180.
If a challenge is partially completed when a level is finished, the progress IS NOT saved unless it involves finding Hidden Spiders. This means the challenges have to be completed in one run through the level. If you die or restart from a checkpoint, you lose any challenge progress since the last checkpoint, EXCEPT completing them. Once a challenge is completed, it stays completed regardless.
Every level has a challenge for finding all hidden spiders, for finding a certain number of spider emblems, and for meeting some criteria while fighting the boss. Even if a challenge isn't "unlocked" by completing another challenge connected to it in the Web, you can still complete it and progress is still counted toward it.
The ones involving timed challenges are fairly generous, and you have trouble with them they can be completed on any difficulty. The Web is independent of difficulty level, so it is possible to complete a majority on the first playthrough and the rest on an easier difficulty. Most challenges that require doing some action against X number of enemies can be repeated on the same enemy (hitting with water barrels, throwing benches, etc.) as long as the enemy is still alive.
Notes on a few of the less intuitive ones:
Hammerhead - Make him destroy 6 columns in the first fight
This can be done easily before even damaging him. He will fire in between columns at the last place he saw you. If you let him see you near a column then strafe across the column so he sees you on the other side of it, he will start to rotate and hit the column with bullets. Move back and forth like that repeatedly to make him destroy one, then do the same for 5 more.
Hobgoblin - Force him to drop 3 bombs
This requires hitting Hobgoblin with either a thrown object or web shots (which is the easier method) while he is priming a bomb to throw.
Hobgoblin - Divert 3 missiles into enemies
When a homing missile is fired at you, go into Accelerate Vision to slow down the missile. Then, run behind another enemy to cause the missile to try to home on in you, but hit the enemy instead
Sandman - Find Juggernaut's Helmet during the final fight
The Helmet stands out fairly well but there is a ton of debris present. I found the easiest way to get this was to look around briefly, and if I couldn't see it, I restarted the checkpoint. After a couple tries I saw it almost immediately and was able to quickly Zip to it.
Vulture - Force him to drop 3 molotovs
Same as forcing Hobgoblin to drop his bombs aside from being able to Zip Kick him as well.
Deadpool - Hit 5 enemies with Bop Baps
You will see a lot of these in the first fight against Deadpool. They are actual Bop Bags with Deadpool on them scattered around the area, and explode either when you hit them or get too close. The easiest way to get 5 hits with them is to stand next to one while a Huge Fanboy walks near you, explode it to hit both yourself and the Fanboy, and do that 4 more times.
Juggernaut - Throw or kick 5 enemies into other enemies
I couldn't find a reliable way to kick enemies into each other, but there is a move (I think called Web Spin) that lets you grab an enemy up close with B then and twirl and thrown them by pressing B again. Many times it wouldn't register me throwing light enemies into each other, but throwing them into the Captains worked every time.
Doctor Ocktopus - Hit 5 enemies with one charge
This is much easier with a Lv2 or Lv3 charge attack for 2099. Keep in mind that if a shielded enemy is nearby, they will stop your attack completely. Wait until you are attacked by a bunch of enemies (like in the reactor rooms) kill any shielded enemies first, then charge attack into the rest.
Carnage - Throw Carnage into the reactor 5 times
This is done in the first fight. He will stand on the reactor a couple times after throwing red goop on one level of the arena, after which you can Zip Kick him into the reactor, but he won't do it on his own enough to get 5 times. You will need to bait him up there by Zip Kicking on the reactor and wait for him to try to zip across to knock you off. Quickly jump off, turn around in mid air, and Zip Kick the reactor again and you can knock him into it. Make sure to do this quickly as each time he will lose health.
Carnage - Burn 25 symbiotic enemies
This cannot be done on the zombies. Partway through the level you will be attacked by symbiote creatures in a room with flaming debris everywhere. Simply grab pieces of debris and throw them, and you will likely hit multiple enemies per throw.
Carnage - Simultaneously ignite 5 enemies
This one can be a bit annoying. In the open areas where Slayers and zombies are fighting, there are red barrels everywhere which will ignite enemies. Let a bunch of zombies gather around a Slayer, then grab a red barrel and try to throw it at the Slayer. It should detonate and hit the zombies all gathered around it as well.