ARAM aka Fiesta Mode

ARAM, otherwise known as All Random All Mid is a recent map addition to the game however the type is not new by a longshot. ARAM has been played since beta and is one of the more popular variants of custom games along with ARAB (Dominion version, all bottom) and Teemo hunting (capture the teemo, etc). 

ARAM works on a few principles, listed here:

All random:
Done in lobby settings, everyone randoms a champion that they own or is currently free.

No Backing:
There is no back function, running back does little for you too as the fountain will not heal you.

Limited buying:
Buying is done at the beginning of the game and while you are dead. The moment you leave the summoner platform buying is restricted until your next death.

Item restrictions:
Certain items were considered ARAM ruining, as such they are restricted. No stacking items or GA. Some Summoners rift items are replaced with the dominion variants (sanguine blade, kitaes razers).

The goal of ARAM is to push down to the nexus and win, just like a real game. However due to the very limited xp gain and gold available, everything gets ridiculous really fast and the game becomes a teamfight bloodbath.

If you are unlucky:

Your team will random no range/poke champions and you will be forced to hard engage every fight.

If you are lucky:

You will poke all day and pretty much enjoy a free win if you can control the teamfights and initiations.

 

Some winning strategies:

Runes - have two basic rune pages for AD and AP, use the third as a tank rune page with GP5 in it (you'll see why later)

Masteries - you can change on the fly so they don't matter too much, just make sure you have a good one

Buying - If you are going to be a major killer or minion wave clear then you would consider starting boots and 2x dorans. Some special cases exist (kennen, vlad, etc - start gun or something gimmicky)

If you are going to be supportive (utility) or initiator/damage soak (tank/cc) you would consider going the GP5 route of Philo, HoG and avarice/kages. This will boost your income dramatically and you can just sit in brush or on tower waiting for a good moment to go in/punish.

How to play your team comp:

This is just as important as itemization. Consider the strengths and weaknesses of the team you have and the enemy team. If you have a hard engage team of Rammus, Tryndamere, Vayne, Galio and Karthus you will want to pick your fights carefully otherwise the enemy team will eat you alive. However if you have a pokey team of Nidalee, Ezreal, Lux with Sion and lets say Shaco/Eve, you will want to poke the enemies really low and assassinate them if they stray from tower. Half or ARAM is the control you can exert from the power of your comp.

Dont be afraid to sit at tower until 6 too, some champs are much stronger late and have a weak early game. 

Some quick things to note:
If you control the brush, you control the lane up to the enemy tower. Vision is key because there are no wards so brush will give you a huge advantage. 

Do not underestimate regen items, half the time you will be doing not very much and regen will keep you healthy during the downtime. Same goes for mana regen, even though it is boosted you might want to consider a chalice or holy grail as early on it will provide lots of mana, ap, cdr and magic resist. If you need to buy try to sneak around and get executed by a tower, remember to leave 10s after getting hit by an enemy to prevent the game from awarding them with the kill.

Most importantly of all, the game is called ALL RANDOM. Understand that you could and will at some point random your worst, most hated, under powered champion. The point of ARAM is to play random champions, take the good with the bad as it will make you a stronger player if you can overcome your weaknesses. 

Remember, ARAM is for fun not for tryharding all day ery day. Just try to have a good, fun time.