There are two parts to the metagame. Lane comps, and team comp. Team comp is basically planning for late game, lane comp is figuring out the safest and fastest way to get there.
Note that when I use terms in lane like Assassin, I make no difference between AD assassins like Talon or Khazix, or AP assassins like Leblanc and Katarina. This goes for the other roles as well- I roll tanky AP in with AD bruisers, for instance.
Solo Lanes get levels faster, and thus get people who scale off abilities. (or just have a really good ultimate) In the current meta, this tends to go to assasins, bruisers or utility mages. (Though some mages stretch "utility", like Karthus's "Press R to gank every lane")
Duo lanes are forced to share experience and gold. With the way gold values work, though, it's generally best if the "share" works out as close to 100%/0% as possible, which makes one character get gold like a solo lane, but less levels, and the other character basically gets the shaft. "Supporting" in lane means you avoid taking your partner's gold, while doing everything you can to make things easy for your partner, AND/OR difficult for the enemy. This can include warding/CV, tanking/healing, and harassing or even zoning the enemy out of gold and experience. They have to be able to do this on what is basically a shoestring, which means you will see a lot of early game champs and utility/healing mages filling this role. their partner, however, needs to be able to focus on gaining as much gold as possible to fill out his build, since he will be behind any solo lane in levels. However, enemy champions give gold and experience as well- balancing trading/harassing with last hitting is part of the skill of the duo lane. This also leads to "Kill lanes", lanes that are primarily intended to kill enemy champions for gold- but even in the kill lane, the best way to divide gold from minions is still "everything to the carry"
Mid lane has the shortest distance between the center of the lane and the towers, making it easier for people with less mobility to react to a gank in time to get back to the saftey of the tower. Being broad and short makes this the safest lane, and while having a duo lane is possible, the enemy can always farm under their tower- barring a team that can dive or harass a turtling character, a solo lane mid will usually beat a duo lane mid.
Top and bottom lane are both longer and narrower, giving bushes on both sides of the lane to sneak into range for harass and abilities. These tend to be more active lanes, and there is more room to zone an enemy safely- They gain a distinct benefit from having two characters vs a single. Bottom lane is also closer to Dragon, an early game objective that grants as much gold as killing 2 enemy champions- controlling the Dragon puts your team in a better position.
The jungle is a wildcard- not a true "Lane," it can be treated as one by characters capable of surviving the first PvE clear, gathering powerful buffs, but generally not matching the gold or experience of a full lane. Their main advantage is the ability to attack other lanes, getting kills or assists (and giving the teammates in lane kills or assists) and forcing the enemy to not be in lane (whether dead, or just going back to heal) allowing the laning ally to gain an advantage over their immediate match-up.
The role of the jungle can be broken up, however, if the team is coordinated. As an extra source of gold, a team without a jungler can push their lane, then while the enemy is dealing with the minions, go into the jungle and kill the monsters, taking gold, experience, and buffs back into the lane. And when the lane is pushed, a fast enough champion can attack other lanes and get back before their own lane is in danger again. Even with a jungler, it is often expected that the jungler will give the Blue Buff to a teammate after the first time he clears it.
If you treat the jungle as a lane, however, that means that, in a 5v5 game, there are 4 places to go to get gold and experience. By definition, that means one of the lanes must have 2 characters in it.
As mentioned earlier, Mid is unsuited to a duo lane because of it's dimensions, barring a champion-specific laning strategy, which is not something the "meta" gets built around. A Double Champion Jungle is possible, but already gives less gold an experience than a real lane- dividing it further is a poor decision EXCEPT when the enemy doesn't use their jungle at all. That leaves top and bottom as the two lanes best suited for a duo lane- and both really are suited for it. Bottom is generally given the extra character because of proximity to the Dragon, but a duo top that knows what they're doing (zoning, freezing the lane, harassing and warding) will dominate a solo top unless the jungler camps the lane... effectively making top a 2v2 lane.
In short, the "best" lane comp with current mechanics is someone who scales well with gold and levels mid and top with top getting the one with better durability or escapes (assassins, bruisers, and utility) with someone who can survive it in the jungle (assassin, bruiser, or tank, though there are exceptions) and a combination of someone who scales better with items (generally a lategame character, such as a ranged ADC) as "carry" and someone who DOESNT scale well on items as a "support" in bottom lane near dragon.
But just because Leona makes a good support, Amumu makes a good jungler, Galio makes a good mid and Shen makes a good top, doesn't mean you should make a team with all four of them on it- Team Fight Composition is just as important as Lane Composition. Riot has been attempting to balance the game around balanced comps for several years, and barring problems with tanky DPS builds (Season 1-2 Meta-golems, and early Season 3 Black Cleaver) they have been largely sucessful. This means that in general, you want roughly equal physical and magical damage, a good mix of CC, and teamfight initiation.
Most of this is common sense, but it bleeds back into lane composition- 2 bruisers bottom will dominate ADC/Healer bottom, but unless the team with 2 bruisers has some good lategame champs elsewhere, they're going to regret it around 40 minutes in, if the ADC is cautious and farms under her tower without feeding. And wherever those lategame characters go, they have to be able to survive the early game without being denied just as hard as the ADC is being denied bot. The popular "Bruiser Top, AP mid, ADC/support bot" has proven over time to be the best general answer to the lane comp vs team comp problem, but it also gives alternative lanecomps a baseline- the recent trend toward assassin mids, for instance, as a counter to the utility based mages that go there by default. But those assasins could be countered by bruisers, who could be worn down by mages or DPS... the permutations are endless.
The Meta exists because it has proven over time it has a very slightly higher win chance, against all other possible team comps averaged together. For this reason it's popular in blind pick,when you COULD be against some random comp of Champs-People-Are-Good-With. But the thing about any dominant stratagy is that it can still be countered- To quote GI Joe, Knowing (what you are probably up against) is (at least!) Half the Battle. I suggest checking out http://penny-arcade.com/patv/episode/perfect-imbalance for more about dynamic systems caused when one play style is SLIGHTLY more powerful than others.