A hatter is someone who makes or sells hats. It is an old trade-name, like baker or potter: the -er marks the person whose work is hats. You still meet it in history, in shop names, and in stories. The famous saying mad as a hatter comes from a real, ugly fact: hat-makers once used a mercury mix that could damage the nerves, so some hatters shook and spoke strangely. Alice in Wonderland’s Mad Hatter plays on that saying. In ordinary modern talk we more often say hat maker or the person at the hat shop — hatter sounds a little old or story-like, but it is still the right word for the trade. (There is an old dialect verb hatter meaning “to tangle,” from a different root; that is not this job.)