Packs more punch than a cozy or soft-boiled but is not too tough or graphic in tone and description. Think Nevada Barr, Dana Stabenow. Usually, but not always, amateur sleuth.
Police Procedural / PI
Think cozy with a bit of an edge, some violence, and maybe some bad language. Often humor off-sets the danger. Think Janet Evanovich, Sue Ann Jaffarian, Jess Lourey.
Some say that any story not set in LA or NY is a regional. Typically, stories that take the reader some place, stories where the location becomes a sort of character, stories where the location is a large part of the hook are called regionals. William Kent Krueger is good example of a regional.
Suspense & Thriller
A Suspense mystery (often called a "Thriller") is a mystery that creates a feeling of tension and uncertainty. When you're reading a book with this kind of tension, it's tough not to feel intensely interested in the outcome, and feel a kinship with the main characters. These are "page-turner" books that keep you up all night in a constant state breathtaking entertainment, furiously reading to find out what happens next.
Traditional & Cozy
Traditionally, a cozy was not just a lighter mystery without blood, gore, or sex with an amateur sleuth. In traditional cozies the rules of fair play are observed. That is, the reader is given all the clues needed to solve a mystery before the end of the book. Readers today are more likely looking for a nice, entertaining, clean story rather than an intellectual exercise, so the rules of "fair play" are considerably relaxed. Agatha Christie's Miss Marple and Jesica Fletcher from Murder She Wrote are examples are cozy sleuths.
A book in this genre features a mystery in which food is well-integrated into the storyline, either peripherally or directly into the plot. The book might take place in a restaurant, feature recipes, or just discuss food in a consistent manner that is vital to the story.
Historical mysteries are set in some time other than the present. Beyond that, they can be cozy or soft-boiled. It's rare to see medium- or hard-boiled historical. Think Ann Perry, Elizabeth Peters, Deborah Woodworth, M. J. Zellnik (Murder at the Portland Variety).