On-hit and attack speed
One of the easiest synergies to work with is the relationship between on-hit effects and attack speed. It is a well-known fact that attack speed is really good on Vayne, as she can apply her Silver Bolts more rapidly. This doesn't mean that other carries like Ashe or Tristana don't benefit a lot from attack speed, but Vayne gets a special boost for building more attack speed that is fairly unique to her. Other similar characters include Caitlyn (to build up Headshot), Twitch, Varus, Master Yi, and Kennen -- i.e., anyone who builds stacks of a buff (or stacking debuff) when he or she attacks gets a bonus from attack speed.
Characters like Kog'Maw, who deals a percentage of the enemy's health as damage with his Bio-Arcane Barrage buff, also love extra attack speed. Characters that deal a on-hit flat effect like Jax, Sion, or Dr. Mundo are great beneficiaries of attack speed buffs. Other characters that build resources on hit, such as Shyvana and Tryndamere, also really love attack speed buffs.
Of course, this means that these characters work great when supported by characters that give attack speed bonuses. Nunu, Gangplank, and Sivir are a few examples of champions that can buff their allies' attack speed (as well as their own). It shouldn't be your only consideration when selecting a character, but in blind pick, it can be very helpful to work toward this synergy when you don't know much about the enemy composition.
The same goes in reverse, though. If your team does not have anyone who really benefits from attack speed, you may want to pick a different character. For instance, Ezreal (assuming your ally is building him as a carry) doesn't really need attack speed buffs as much as he needs flat attack damage. Graves can get by without attack speed buffs due to his many damaging skills, though he relies on his basic attacks more than Ezreal does.
Snares, slows, and repositioning abilities
Snares and slows are crippling to melee opponents that have to close the distance with the enemy. Although not every melee foe is created differently (Riven will laugh at most slows in a lot of cases), snares and slows can be generally very painful for a melee opponent to fight against.
Put in reverse, though, characters that need to close the distance or rely more on positioning love these types of abilities when they're used by allies. The red buff on Summoner's Rift is especially valuable for melee characters (though it's great for ranged characters that can use it to kite) because enemies cannot flee their attacks.
Characters that use positioning as a key element in combat also love anything that inhibits enemy movement. Vayne and Poppy can get in a good position to wallslam the enemy much more easily when the enemy can't move as fast as they can. Both characters naturally have abilities to help them do this, but it never hurts to have more.
Anyone that has a skillshot also benefits from slows, though someone very accustomed to leading the target may accidentally lead a slowed opponent too much. It is a lot easier to tag someone with a Nidalee spear if he isn't moving.
Some characters, such as Singed, Urgot, Blitzcrank, and Volibear, can actually reposition the enemy in battle. Characters with taunts, such as Rammus, Shen, and Ahri, also can do this, though in a more limited way. Pulling foes into key AoE attacks or dragging them deeper into your team's ambushes is also very helpful. Poppy, who can use slows to help get better angles on her charge, can also return the favor by charging behind the enemy and slamming him back into her team.
Repositioning abilities are very important to coordinate with other allies. When playing Poppy, I often initiate by charging an enemy laterally into a wall. I can't count the times when I've accidentally hit a foe out of the way of a Lux or Ezreal ultimate fired by an ally just as I engage. It's important to let people know what you're planning with repositioning abilities. As Singed, I've "accidentally" thrown someone out of Nunu's Absolute Zero (actually I was purposefully stealing the kill from him), which is definitely not something you want to do (unless you're a jerk).
As a small aside, melee champions (or any champion) can use their bodies to block enemy movement. If you put your character in front of the way the enemy must run to get away from your team, your foe must run around you to escape. This will give you only a little bit of advantage, but it is a good trick to know, especially if there are some of these abilities on your side. Sejuani can slow the opponent and run around him, making him run around her. Then, she can just run past him and and repeat the process, giving her ranged allies a lot of extra time to deal damage.
Covering for weaknesses
Some characters have big limitations, and a good team comp should help address those things. Aside from her ultimate and her fairly weak slows, Ashe has no real escape if she is ganked. A support with hard stuns like Taric or Alistar can help her overcome this weakness and allow her to lane more safely.
Katarina and Nunu have powerful ultimate attacks that can be interrupted. A quick silence, stun, or knock ability shuts their powerful attacks down and puts them on cooldown. A teammate with an area stun, knockup, or taunt can dramatically improve the chances of a successful ultimate by either of these characters. The classic synergy is combining Galio's taunting ultimate with Katarina's, but there are plenty of variants. Kennen and Morgana's AoE ultimates also benefit a lot from AoE stuns or snares. Combining Morgana and Amumu's ultimates is a recipe for pain, and the stun from Morgana's ult can even lead into a Katarina ultimate for even more punishment.
As I mentioned above, essential skillshots can be made guaranteed with a stun or snare. Ahri's charm is a bit of a gamble when thrown from a safe distance, but against a stunned foe, it is a sure thing that keeps the stunned enemy out of the fight for even longer. It's important to time your skillshot CC to hit as close to the expiration of your ally's CC as possible in order to get the maximum effect, but don't throw it so late that the enemy has a chance to dodge.
Play to your team's strengths
Above all, be creative. Don't fling people out of Nunu ults as Singed, and if Poppy is initiating, don't throw skillshots until after she's charged someone. The real key here is to play for the team's victory, not to pad your personal kill score. Yorick and Kayle can really turn a fight around with clever ultimate use on the right ally -- too many players use their ults only on themselves.
If Teemo is spamming mushrooms in the jungle, bait people into them before you turn around. I've seen many an ally completely ignore friendly traps and dive headlong into a 1v1, which often results in the ally's death. If the enemy runs into a Shaco box beforehand, the 1v1 becomes a lot more of a sure thing even if Shaco is nowhere nearby. These characters thrive on allies; chasing Nidalee through the jungle is almost always a bait, but chasing an out-of-mana Kog'Maw through the jungle can trigger just as many traps if he's paying attention to where they are.
Above all, don't instalock a champion. If you'd like to play someone, select the character you want and be prepared to change as needed. It also goes without saying, but don't dodge the game because you couldn't play Varus. That's not being a team player.
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