The Reverend Samuel Parris is very concerned about his "name." By that, we mean that he is very worried about his reputation -- the way people think of him.
This is partly because he is the town's minister. In Puritan times, this was a very important position and he would have felt the need to have everyone trust him. It is also partly because of his own ego. He really wants everyone to look up to him. This is why he demands things like golden candlesticks for the altar. This is also why he is so worried about Betty and her apparent connection to witchcraft.