In what ways can the assassins be useful, and what is their job in a team fight and the entirety of the game?
Assassins carry the roll of killing off the weak targets like support and carries. During laning phase they can roam around abit picking of the easy gankable targets. In team fights there primary focus is taking down lik said before, the carries and the support. Mostly circumventing the tank and going straight for them so that the rest of the team has an easyer time to take the rest down, because their carry is either fighting for their life or dead.
An assassin is typically a chamion with high mobility and burst potential, but limited sustained damage. Generally, they counter the ad/ap carries quite significantly, as they can move around them quite quickly and burst them down.
During the laning phase the assassins are typically in either the top or mid lane, depending on who they enemy picks. During this phase it's a good idea to harass the enemy as much as possible using your burst/movement speed to your advantage to win trades. Romaing to help other lanes is usually a good idea as well, just as long as you dont leave your lane for too long to do it.
During mid-late game, your job is to kill the carries quickly. Focus on the squishy targets, using your mobility to get around any CC.
Simply answer: Kill the AD Carry, AP Carry and Support, they can even kill the Tank, but the point of the assassin champions is to nullify the opponent damage/support, and get advantage for your team in team fights.
Here is a list of the Assassin champions in LoL.
but the most common are the one who has the flag of assassin, those are:
What is characteristic in those champions are the high burst damage (in 1 or 2 seconds they can finish a full health champion easily), they high mobility and they possibility of good escape/survivability.
They are usually solo top and mid, so they can farm fast and do they job.
To nullify a assassin is hard, but basically you need to avoid his farm, and force ganks on them.
There is not such thing as the best assassin, they all do they jobs quite good, but they are more counter-picks.