Your characters, especially the protagonist, are the most important elements of a mystery story. In order to appeal to the reader, they must be believable and dynamic. They are more than just words on a page -- to engage the reader in the mystery at hand, they must function as real people. As with real people, they should have their own personalities, hobbies, fears, and passions. Forgetting this can cause your cast to become flat and detach the reader from the mystery at hand.
You can use mood to enhance the feeling of your story through several mediums. Settings are an important one -- by vividly describing the atmosphere, the reader can put himself into the story and the events be that much more impacting. Creating suspense is another important method in developing an intriguing mystery. Do this by embellishing the moments closest to a climax. Time should seem to slow down more and more as the reader gets closer to an exciting moment, allowing them to experience every tiny sound and event. You might include things like rustling curtains, the sound of breathing, and the echo of footsteps.
Without an interesting background, your mystery may seem paltry in comparison with one that includes an absorbing history. Engaging back stories provide a platform on which you can build your storyline in a more stable and realistic manner. Including a good background is also an excellent way to incorporate character information, making them more colorful and interesting.
Be fair to your readers and make sure that the clues are something that someone with a sharp eye can pick up on. Don't make them obvious enough that they jump out and spoil the surprise -- this is a mystery, after all! A good mystery only needs one or two important clues buried within the plot. This is enough to make the pieces fit together, but not so much that they give things away too far in advance. A few red herrings -- or false leads -- can be another good way to add fun and mystery to your plot.
The conclusion is the "ah-hah!" ending of your story, where all the pieces will fit together. It is important that you know how your mystery will end before you ever begin writing it. If not, your story can seem disorganized and contain too little or too many details for even the most mindful readers to pick out the pertinent information. The best conclusions are filled with suggestive details, revealing the mystery slow enough for the reader to feel as if they've helped solve it, and fast enough to be exciting.