This is based on the Biblical New Testament story where Thomas, one of the apostles of Christ, refused to believe that Jesus had risen from the grave without seeing evidence. On seeing this evidence, he praised those who believed without proof.
Typically this phrase is used in a midly derogatory. It describes someone who refuse to simply believe in something out of pure faith without asking for evidence. A devout Christian might consider me to be a doubting thomas because I don't believe there is a God since there's no evidence. The Christian considers it a bad thing that I'm not simply willing to have some faith.
Fanatic: How can you not believe in God? You don't want to be a doubting thomas?
Athiest: Normal people don't consider it a bad thing when others ask for proof when they make extraordinary claims. If I said that I am Zrlak, King of Mars, I would have little respect for you if you believed me without asking for evidence yet you expect me to just believe your equally ridiculous claims?
A phenomenon where someone believes they haven't taken a hit, because the draw was so smooth; resulting in the smoker performing the Doubting Dragon, where smoke billows from their nostrils and mouth whilst voicing these concerns.
*Takes massive hit*
Smoker: "Oh no, I didn't get anything!? Did it work??"
*Meanwhile, smoke is billowing out of their nostrils and mouth like a Doubting Dragon *