That's right ladies and gents, assassins. Cold blooded murderers stalking around the fields of justice practicing their 
blood soaked art.
So assassins, what makes them strong?
Why pick them over another playstyle?
Why are they in every game?
All that and more I will attempt to answer, with as much wit, sarcasm, and typical teenage humor as I can handle.
To start this out, let us discuss the meta. The meta game is the "right" way to do things, at least by way of watching what the pros do. An ever changing beast, the meta will vary from season to season based on changes to champions, items, or gameplay changes in general. Season Three the assassins descended into the rift and began making the play into a shadowy nightmare, where the wrong turn, or a misplaced step would land you six feet under.
Assassin types:
Starting off you have your high mobility assassins, 
Shadow Dancer, and 
The Shadow Hidden in the Dance. These assassins just beast around the map at mach twelve and cut the throat of anyone that was previously breathing, don't worry 
Ghost Rider you are safe, you too 
Poor Yorick (we knew him well).
Next up you have the stealthers, 
Sadism and Masochism, 
Bird Foot, and Yeah, the dance continues. They wait in the shadows, then the moment your pinky toe crosses the line, they jump in and tear out your heart Mortal Kombat style.
Last, but not least, the hyper carries. 
The Last Samurai, 
Windamere, and 
Made in China.
All of these assassins are extremely bothersome to deal with, and have the habit of jumping in when you least expect it to finish you off, then get fed and 
snowball out of control. So what do you do against them? I'll get to that later, I asked other questions before, hold your 
Horses.
So why would you pick assassins instead of say, bruisers. Well, Assassins have a very clean playstyle, they build as much damage as possible, they jump in, wreck shop, and either die immediately, or get out. Bruisers on the other hand have to gauge much more, they have to be aware of their health, the fact that their damage is over a period of time and not instant burst, you must judge how your damage stacks up against the enemy's. You have to be able to off tank, while still doing decent damage, and have very precise management of your skills to maximise damage output while not dying in the process. Assassins on the other hand can hit all of their skills in a certain order and someone on the enemy team will more than likely be dead.
Assassins are not explicitly better, in fact most bruisers can fight assassins and typically win if they are both doing evenly, but the real danger of assassins is that they will 
explode your back line of carries, leaving your team's damage severely lacking, and turning the fight in their favor.
Now for why they are in EVERY game, even in competitive play. Assassins accelerate the pace of the game, it makes it exciting, it makes you think, it makes things risky. Assassins excel at getting a few kills and then roaming around, killing people in other lanes and helping the other lanes get ahead as well. An assassin killing the enemy mid laner opens an opportunity to push and farm safely. The same goes for all the other lanes, and even the jungle. Safety from invades, buff and monster camp steals, and getting attacked and killed in the jungle allows your jungler to safely farm up, apply pressure where needed, and secure objectives.
Back now, to the "what do you do against them" situation. Most people in lower rankings would say "Just kill the shit out of them" or "Run like the devil himself is behind you". Most people in higher rankings would have an actual strategy. CC them, kill them before they kill you. "You know that someone is out to get you, so wake up an hour early and go kill him first while he is sleeping". Take advantage of their early aggresion when they are stronger than most, bait them into a trap, have your jungler gank them when they are in your area of the map, you must take them out early on or they will be an issue the entire game.
Notice the last few sections are not full of jokes and terrible puns, that is because it is a serious thing that I am saying.
To sum everything up, If 
You See These People 
run for your life or 
come back once you've collected all the bits of yourself.
Alright I'm done now, ciao.
Yours truly, Kaboose.