Synergy basically works like this.
I as a support have a particular set of skills that compliment my ADC's abilities. In turn, my ADC benefits greater from my support abilities than other ADCs do because of said synergy. I'll list a few examples to articulate this point a bit better.
Nami is a CC machine with very little damage and moderate CDs. Her synergy with MF comes from not only their ults having a wonderful synergy, but Nami's Q stun is much easier to land if MF is using her E slow. Furthermore, MF has a passive on her W that adds magic damage to each shot, Nami's E also adds magic damage and slows for 3 attacks after cast. This allows MF to set her E up better and gives better chase.
For Varus, it comes back to the slow field+stun as well as the passive magic damage.
Sivir is a bit different. Her chase passive works well with Nami giving her little speed boosts with every spell and the ability to add a small slow to her AA's makes Sivir's chasing potential amazing.
Once again this comes down to Sivir's passive. Galio has an AoE sow and a line of speed he can throw out. Used together, these allow Sivir to get in close very fast and wreck face. In addition, his bulwark gives additional MR and armor so that Sivir can take a few more shots to the face
Both have amazing burst potential and this is as close to a kill lane as a traditional support can give you. Also, Sona is basically queen ***** of supports and will do well with just about any ADC.
POKE. This is the single most annoying lane ever. You will be hit by a Q from one of them eventually and if you're not it becomes a mini-game of dodging shots so your farming ability is reduced. As both have sight revealing traps as well, it makes zoning harder as well as ganks.
This is one of those lanes that everyone hates to go against. Nunu just sits in the bushes and slows you anytime you try and get close to his carry. He then gives the ADC a speed+AS boost so they can wail on you harder. the reason Vayne and Cait are used a bit more than Kog is that they have a damage steroid that is completely dependent on them achieving a certain number of autoattacks. Kog is used because he naturally gets an AS boost from his Q's passive and his bio-arcane barrage allows stacking AS on Kog to actually be a viable build. As Nunu is capable of providing this for him, it allows Koggles to optimize for more damage as well as have a tank to protect him versus a squishier support.
These are a few examples and if you'd like a few more I can provide them, but this should give you a basic overview of synergy.