I am no expert, but from what I understand, Neo-Druidry is the modern practice of the Druidical belief system. It is not the same as what Druidry was a few hundred years ago, but pretty similar. It isn't so much a religion as a reverence for nature and harmony. Back in the older days, the druids were healers, advisors, and bards who recounted history through song and poem. Since very little is known about the older druids due to extremely little literary history. The modern day druids are referred to as Neo-Druids, referring to the fact that they are reviving an old belief system based on the idea of it.
It would be near impossible to be just like a druid from the old days, although I am sure there are probably small groups out there who are following the same practices as back then with teachings being passed through oral history.
Well as SolonKing said Neo-Druidry is actually a relatively modern practice and so the odds are it's not buried anywhere in your ancestry although it is possible that it calls to you for another reason. Druids were indeed a priestly order at one time although there are no remaining practitioners now days and due to a lack of historic evidence we know virtually nothing of them, including whether they even had children so the odds are that's not really in your ancestry either.
That said it's important to note that druids were an order of priests and thus "druidry" was never actually a religion on its own just as we have monks today but monk-hood is not its own religion. Whilst it is impossible to claim to have any Druids in the family tree (as there are no records of it) it is still possible to claim Celtic ancestry. (as a side note and little correction druids were not bards, given our current understanding bards were an entirely separate order more akin to poets than priests).
Whilst it is unlikely to have a particularly long Neo-Druid heritage you may have one or two practitioners a couple of generations back. If you're interested in Neo-Druidry there are a number of fairly good resources out on the internet including adf which is a good resource to understand neo-druidry itself, and the grove of Dana or druid circle which whilst still neo-druid focused also incorporates a little more of Celtic lore (see below for links). Ultimately neo-druidry is a fairly open structured religion and many followers have their own take on it so it can be a little confusing at first but if it interests you give it a read over.