In The Crucible, what does the court accept as evidence that someone is a witch, and which characters seem to consider this evidence valid?

The judges in the play and in history chose to accept "spectral" evidence in cases of witchcraft.  While spectral evidence would never hold up in today's high-tech cases of DNA, reconstructive crime, and video footage, in Salem's day the religious nature of and neighbor infighting connected to the trials caused the judges to set aside logic.

Spectral evidence is primarily when a "witch" sends  his/her spirit to harm another or to cause mischief. Abigail's scheme with the poppet to show that Elizabeth Proctor wanted to harm her, and Abigail and the girls' sudden yellow bird sighting (Mary Warren's supposed spirit) in the courtroom serve as specific examples of such evidence in the play.

Many of the common townspeople seem to buy into the veracity of the evidence because most do not speak up against it and their uneducated state forces them to trust solely in the authority of the judges and religious leaders.  Judge Hathorne at times appears to find the evidence credible, as does Rev. Parris (perhaps because he needs so badly for the girls to be right). Judge Danforth simply uses the evidence to his advantage, because he is too observant and self-serving a character to actually believe that the evidence is valid.  He obviously sees the silliness of it but has gotten himself so embroiled in the trials, that he does nothing to disallow spectral "proof." Rev. Hale also serves as one who becomes thoroughly disillusioned with the court and its use of "voodoo" evidence because he knows Abigail's true motive and doubts Rev. Parris's sincerity.