Positioning? What's the right way to go about it?

Let's look at this from a team fight perspective and expand from there.

The assumption here is that you have a balanced team with a tank, rdps, mage, fighter and support.

The tank and fighter are going to be what we call "Front Liners". They are going to be the closest champs to the enemy's champs.

Then comes support. The support champ should generally be in range of all the allied champions at the same time if possible. This lets the support champ throw heals, shields, other buffs on his/her allies.

The "Back Line" will consist of your mage and rdps. These guys should be farthest or least accessible to the enemy champions as you dont want your dps and mage (main damage dealers) going down too soon in the fight.

Now lets go over the roles. Front Liners are going to soak up the initial harassment damage in a team fight and maybe even an ult or two. To put it simply, if your tank is taking damage they are doing their job and winning you the game, even if they die. The fighter is going to be dealing some damage taking some damage but should generally be comfortable with the support of their allies.

Support needs to be wherever they are needed. This might be the most complicated role to play. You need to be in a place that is both safe but advantageous. If you can't buff your allies you aren't doing your job so dont be affraid to be in the mix a little bit but watch for assassins and ults like malzahar's, warwick's and ashe's that will leave you out of the fight for some time and probably dead.

Back Liners are your dps characters. Physical, magic mage or whatever, they are your squishies and glass cannons. I am refering to them as back liners not because you want to form physical lines on the battlefield but you want your dps characters to be the least likely to die. As a dps character you want to be close enough to deal damage and no closer. Also as a dps character it is important to focus down a target once the fight has been initiated but make sure it isnt their tank, remember tanks taking damage and/or dying means they are doing their job.

So to expand this Front Line, Back Line concept to the rest of the game is actually fairly simple. Dps wants to be in a position where they can deal damage but not take a lot of damage. This generally means have allied creeps or champions between you and the enemy target unless you are zoning, which is a more aggressive play. As a dps character you always want to be looking for the opportunity to capitalize on enemy mistakes or allied ganks so that you rack up some kills and get money and be aware of ganks on yourself as you will be the target for the enemy team. Keeping a team's dps characters underleveled and/or underfed tends to have a substancial impact on the outcome of the game.

Outside of team fights tanks still want to be absorbing the damage and setting up the dps characters for kills. Your main priority, even if you die instead, is keep your carries (dps characters generally) alive.

Fighters have a sort of hybrid role outside of team fights. They want to be in the fray dealing and taking damage, so long as you are damaging their carries, because they tend to be good at both. Warwick for example, can run up to an enemy champ and use hungering strike. Sure, he might take some damage going in but taking the health from a champ like Ashe and restoring his own health, he tends to come out on top.

And finally support. Once again, you need to be where you are needed. You want to be helping your carries get kills and keep your tanks alive. An important thing to remember is that you want to keep yourself alive too because if you are dead, you aren't helping anyone.

I think positioning can't be considered a science with hard rules because it is completely situational with the "correct move" not always being obvious. Sometimes more aggressive moves pay off and sometimes they blow up in your face but one learns which moments to capitalize on with experience.

General position is pretty simple, but the major nuances come down to a champ by champ basis with some being dependant on your own lanemate and the other teams laning champs.

Some general rules of thumb though: Ranged champs, unless zoning, should usually keep a healthy barrier of minions between them and the other champs, it keeps the nasties like amumu and blitzcrank from grabbing your face and nomming on your soft and tenders. Bushes are your friend in all cases, it breaks LOS with other champs and makes you untargetable (directly, skill shots don't apply to this) so use those bushes to your advantage, and if you have a surprise champ like jax or mundo or many skillshot ranged champs it makes a wonderful place to attack from and then retreat to.

Always be in position to herd creeps, if they are being worthless auto-attackers, be one too, just don't be worthless, rotationally hit creeps to get them low and always make sure to leave yourself the ability to last hit, the best way to do this if you aren't ranged is to sit near your caster minions giving you ample time to walk up, smack the bejezus out of a creep and then walk back w/o taking too much in the way of pain.

If someone's looking at ganking make sure to set up the lane properly for a successful gank. Let the opposing team push past mid-point for your lane giving you range to beat on them before they hit their tower. Remember if you over push your lane you'll almost never get a good gank off and you are set up to be penetrated by your least favourite jungler. So a good rule of thumb is try and keep the lane pushing back and forth roughly at near the midpoint of the lane. With a well timed gank the lane should easily be yours.

And as always, last hit like it's going out of style, poke at your opponent as you see fit but (especially early game) don't over extend and always make sure you have mana/summoner spells for a clean getaway if you do.

Firstly, I would suggest you check out this question and answer if you haven't already.

More specifically, I think you should at least watch the following video on Zoning, produced by Shurelia, one of the developers at Riot Games (who make League of Legends)