A mystery is a story where the action is driven by a crime or a threat of a crime-- usually murder. A crime occurs and the villain is unknown until the end.
The private eye novel, the amateur-detective, usually someone with an interesting or unusual profession--hard-boiled and cozies. If it's a young adult book, the amateur detective probably would be a young person. There are even some mysteries where the one who solves the crime is an animal, a werewolf, vampire- just about anyone. The police procedural, romantic suspense, historical mysteries-combination of any of the above.
The classic definition of the thriller is when an innocent becomes involved, usually by accident or coincidence, in dangerous events beyond his or her control, or could be a spy novel. Usually the bad guy is known and the hero is either trying to escape from him or catch him.
Suspense is when the hero or heroine is in a constant and increasing state of danger which could be in any mystery.
A woman in jeopardy mystery is what Mary Higgins Clark writes.
Mysteries of an earlier time were more interested in the exterior of the heroes' lives, now more interested in the interior lives. You must create a credible protagonist that will help the reader's disbelief to be suspended. You need to know where the character is emotionally. You should build a dossier on the characters. You have to know the back story of your characters-you may or may not bring in the back story-but don't start with the back story. Bits and pieces should come out-maybe internally. Don't lump it all together in one place. Villains also have issues and should have a back story.
Do your research.
The plot comes alive when you introduce the characters who cause the events and are affected by them. When naming the characters, be careful. If it isn't a cozy, don't give them silly names. Don't have them all begin with the same letter, sound too much alike, or rhyme. Characters should be real people, act like real people.
In a mystery you need a victim and a murderer and a motive. Out of the motive the solution arrives, so you need a sleuth. To begin with you'll need six elements, three main characters, a crime, a motive and a solution.
You must set the scene for the crime--where the action begins. The victim must be in a place where the victim has access and opportunity, the cause, and the means for the murderer to make his/her escape.
The story ends with the criminal being confronted and caught. Invent a criminal who is interesting and driven. Research your criminal's type and invent a background and childhood to match the crime. There is a huge difference between a sociopath and someone who kills out of desperation or passion. The story must provide a satisfactory explanation of the motive, clues and links between characters before it ends. Mysteries don't have to have a happy ending, but all plot ends must be tied and tidied.
Mysteries are realistic. There should be tension and danger or at least a threat of danger hanging over the lives of the characters, including the detective. The criminal must have a serious motive for his evil deeds and exert great effort not to be captured.