You know, I had no idea until you asked, and now that I've googled and Wikipediaed it, I still have no complete idea.
It seems they are something between a professional doctor and a nurse with qualifications that need academic time that falls between a doctor's and a nurse. Apparently they are very common in Russia and around that area which is probably why I've never heard the term before.
From what I gather, I would say it requires all the skills of a nurse and part knowledge of a physicians. But to be honest with you, I wouldn't trust a Feldsher with the same job I'd expect of a Doctor's.