A sense of story is understanding that stories have a beginning, middle, and end. Students with this ability know stories have characters, a setting, a problem and a solution. They know that a story has many events that lead to a climax and then usually wraps up and ends.
So why is this important? When a child has a sense of story they are more easily able to comprehend a new story. They expect the story to follow a certain pattern. They are able to make predicitons to figure out what may be next in the story.
One way to promote a sense of story is through story maps. A story map is a written diagram you fill in for a particular story. I have found a few online that I thoguht you might enjoy using with your student.
For kindergarten and first graders you might want to start with a story structure map. This one simply gives you a place for a beginning, middle, and an end. You can model your thinking process the first few times to fill it in. Then you might have your child dictate to you the answers.
For Elementary students, here is a nice story map from Scholastic. You read a story and then either have your student fill it out, or they can dictate answers to you. If you have never done this before, then I would suggest just concentrating on one or two boxes for your first story. Depending on how your child does, you can add other elements. Another way is for you to fill it out completely modeling your thought process. Talk about the map and fill it in expecting that your child will only listen. I would model several times gradually having them share their answers. How often would I do this? I would not do it for every story you read or they read. I would work on 1-2 times per week.
For older students this is a nice story map from Houghton Mifflin. It is a bit more complex. I would use the same procedure discussed above. I would take it slow and gradually add more of the responsibility to your student. This one is good for older kids as well.
Lastly, I found this interactive story map for all ages from The National Council of Teachers of English and The International Reading Association. Click on the link to get started. Follow the instructions to fill in the parts of the story. I had fun playing around with it.
I am hoping that this gives you a place to start. Developing a sense of story in your child is an effective way to develop reading comphrehension.