The most influential roles, in my opinion, are as follows:
Jungle > AP Carry > Offensive Support > Solo top + Teleport > AD Carry > Defensive Support
The jungle is responsible for controlling the map, and he is the random variable that can bring kills. To do good ganks, know where the enemy puts their wards, and then go around them. 75% of a gank's success is already done there.
AP carries can both carry the game early and late game. It's easy to win your lane if you know how, and if you do, you can push your lane while going to gank others. Ganking other lanes = your lanes are in better condition = win.
Now why do I rate support as more valuable to solo top and AD carry? A crappy support can easily make your bottom lane fail, and that makes only 3/5 of your team effective. If your support can't or doesn't ward, your lane will lose. Aggressive supports (Alistar, Leona, etc) also create plays bottom lane early game to help feed your AD carry. As Bigfatjiji has once said recently in his AMA, the support carries the AD carry early game. AD carrying late game just requires not getting caught too far up and right clicking, so it's SOMETIMES good to leave to others. Late game usually doesn't come for ranked games because of raging anyway.
Solo top just afk farms, so eh.
To win your lane, just know your match-ups. Know match-ups and battles you can win and ones you cannot. If you're Irelia and you're laning against a Garen, know that your ass is going to get kicked if you go offensive, so just stay farming and wait for a jungle gank or something. Know what every champion does and when their ultimates are up or not. League of Legends is ALL about knowing all the factors (what summoner spells the enemy has up, if they have their ultimate, if you're a higher level, what items they have and the strengths/weaknesses from that build, where the enemy jungler probably is, etc.), considering them in risk and reward, and making the right decision from there.
Also, KNOW HOW TO LAST HIT. Even the slightest misses can make you lose out on a lot of gold if they build up. If you miss 1 minion out of every 2 waves (6 minions per wave), then you're losing 10 CS out of 120, and that's ~200 gold lost by 15 minutes or so.