Elise is a good pick for every game. She is strong in the jungle and good for ganking. I would say Elise is the first champion that made me feel very safe, even if the enemy picked a strong counter jungler eg. Lee Sin, Nocturne or Xin Zhao. She can make many plays, especially if you know how to use the rappel well.

Strong points:
Good burst (Q from both forms)
Good CC (Stun)
Good mobility (Movement speed in spider form + movement speed quints)
Good escaping skills (rappel)

Weak points:
Squishy early
Hard to understand/play (Stun is hard to land)
Very weak against CC (Once CCed, you're probably dead. Use rappel well)

Elise can be played defensively too. If you're afraid that you can't reach the enemy carries, let the tanks buy you some time. You can choose to stick with your carry and try to peel, burst any enemy trying to get too close to your carry. Your Q spells can burst down the enemy tanks very quickly. Once the engagers are dead, you can quickly rappel further into the fight to help your team finish it up.

Marks: Magic Pen (No explanation needed)
Seals: Armor (For you to survive in the jungle)
Glyphs: CD Reduction (MR is not needed in the jungle, CD allows you to use your skills faster)
Quints: Movement speed (Why deal small extra damage when you can run, chase & gank faster?)

It is not wrong to use Ability power quints, it's personal preference. But you'll feel a significant difference with extra mobility.

Why 12/18/0 and not 18/12/0? Both are acceptable, but if you actually do some calculations, the Cunning tree actually provides a better burst damage.

ThunderLord's Decree does a high average damage throughout the game than compared to Deathfire's Touch (Elise doesn't need much AP. Deathfire works better with champions buying more AD/AP and in long battles. Thunderlord scales with level and works in short fights, which makes it more reliable for bursting.)

Cunning also gives slightly more magic pen, deals more damage to champions below 40% hp and restores HP & MP with kills/assists. Savagery helps to kill jungle camps faster. We do not need tough skin because the spiderlings can take the damage for us.

First camp: Always start Gromp first. Add 1 point in W and start in human form. At 1:53 release a spider at Gromp's spawn area and quickly change to spider form. Once Gromp spawns, smite it right away and activate W. Make sure your leasher don't hit Gromp too much or they might get a portion of the experience, or even worse the kill. You can use your spiderlings to tank some of the damage. They die after getting hit by Gromp once, but it's alright because you'll spawn them again.

Second camp: Add your Q now. You can start to consume your first red pot. Using your human form, throw a Q or auto attack once, then use W to get a hit on all 3 monsters. Swap back to spider form when available and activate W and Q on the Blue Sentinels. Also use your spiderlings to tank a few hits after your W expires (Don't let them die too early because you cannot spawn them that quickly). Finish up the smaller golems without using your W.

Third camp: Do not smite the wolves. Start with human form again, use your Q and W to spawn your spiderlings. Change to spider form when available and quickly finish the wolves.

Fourth camp: Go directly to Red Brambleback (Smiting Red Brambleback will give you HP). Again using the same combo, spawn your spiderlings. At the same time ward the brush beside you (in case someone is waiting to invade). Change to spider form and attack the Red Brambleback. Once it is low on HP, change back to human form, spam your spells quickly and smite it once the HP hits below 400 (Good practice to prevent your buffs from getting stolen). You can choose to go to the fifth camp or gank which ever lane is nearer (Top or Bot) If you see no potential ganks, continue the jungle camps. Bare in mind you still have 1 red pot left. I recommend you to use it for ganking (Safer to gank with near full HP).

Standard combo: In human form Land your stun, W , Q, Change to spider form, W, Q. Continue auto attacking your target to finish up.

Your E is the hardest skill to use. Many new players don't understand how Elise's rappel works. There are 2 types of rappels, 1: On a target & 2: On the ground. The difference: On a target, you quickly rappel up and quickly rappel down. This is for dodging quickly and continuing your assault on your enemy. There is a bonus attack speed for your spiderlings if you use this method. On the ground, you rappel up for a maximum of 2 seconds. You can choose who or where you want to land at. This is a good way to dodge skills, avoid death and also escape (Going near a jungle camp can help you rappel over a wall).

For ganks: Stun is important.
The stun is long ranged, but fast. You have to predict well and land it accordingly.

The trick to landing your stun: Build pressure.

How to build pressure? Simple. You don't use your stun first. I know.. If you don't use your stun first, the enemy might back off or flash away to escape. But if they flash off without you even trying any tricks, wouldn't it be great? Build pressure by using W first (especially if the enemy tries to dodge left and right), then you throw your stun. Why? If you use your W first, the enemy will focus on trying to dodge or evade the damage from the spider and they'll pay less attention to your stun. This doesn't always work, but it helped me land my stun more than you think.

On certain occasions, the enemy might panic. They might just burn their flash even before you throw anything at them. At this point judge if they are reachable. If they are, a flash stun will finish up the gank.