Can someone explain how to tell certain bot lane synergy?
the only example i can think of is that people say cait / nunu
and vayne / leona.

so how can i know which support compliments which adc??

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+MF/Varus/Sivir:

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.

Galio/Sivir

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

Sona/Graves

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.

Nidalee/Caitlyn

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.

Nunu+Caitlyn/Vayne/Kogmaw
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.

It's also about the Adc's playstyle, like if they prefer to be more aggressive, then a support like Leona would work well with them, due to her hard cc and dive, but then supports like soraka pervade a passive playstyle, with her heals allowing the adc to sit back and farm

I just want to say that these posts are good. I usually am very aggressive and play supports like leona or sona, but when I play with my friend who is extremely passive I hate it and playing leona turns into a huge waste because the adc will not be ready for the engage, even though you are leona and all you do is engage. So then if you have a passive adc play soraka, taric, sona, or alistar. generally people with sustain help out the passive players and some cc can be good to ensure that your adc doesn't get killed. example is stunning ezreal so he can't do his burst on your adc, thus you just saved your adc a little bit of health.

Ashe+Nami is another potent combo.

Ashe's slows will help Nami land her Q. Plus, Nami's E proc stacks on top of Ashe's slows for even more slowing. And Nami's W will help sustain Ashe through her poor trading phase early game.

Late game, Nami's Q and R will protect Ashe and allow for disengaging or repositioning as needed.