Lulu is an aggressive support who excels at making engages one-sided through shields and cc, with low mana costs to keep constant engages happening. I don't feel like she outright counters any support or adc, but she certainly fairs very well all around as a support with no real hard counters. She fairs a little worse against heavy poke supports like sona and nunu, but she's still pretty safe in those match ups. She plays well with somebody like graves or ashe, and is pretty good against people like mf.
Where she excels is teamfights, as her ult is incredibly powerful as a counter-engage, and she's deceptively efficient at keeping champs alive through strong, low cost shields and making enemies run around as squirrels/cats/cupcakes/dragonlings etc. etc. You play her for her late game, but her early game isn't bad either. She's basically the pocket support of the initiators and tanks, rather than nunu, who's the pocket support of the adc.
As a support, she's not in a bad spot and is still pretty viable, but she's the middle ground between tanky initiator support (blitz, leona, taric, ali) and the pocket support (nunu, janna, sona). I'd prefer a specialized support over an all-around support.