How many books have been praised for being 'page-turners' or 'edge-of-my-seat' reads? There are two elements to capturing that sense of urgency: suspense and action.
Suspense: This can take the form of anything from Hitchcockian levels of anxiety to a simple description of your main character's environment. The important thing to remember is the stakes for your character. The captain of a nuclear submarine can have the same sense of urgency as a student who hopes the class ends before he's called to answer any questions. For each of those characters, the suspense comes from their relationship to the stakes of the scene.
Action: This can be external or internal. In recent books and movies, external appears more popular. It's easier to see a conflict when fists, knives, or bullets are flying. However, an internal conflict can be even more rewarding to write and to read because the outcome is less predictable and the battleground is more concentrated. Use your writing to convey the action, whether it's external or internal, in a way that emphasizes the stakes for your main character.
In the writing prompt below, write a scene with the character I describe. What are the stakes? Who is she fighting? What is the outcome? Is there a cost to her (for winning, or for losing)?
Writing Prompt: With papers clutched tightly in her arms, she started around the corner, only to see the person she was trying to avoid. She ducked back behind the corner and took a few deep breaths. Maybe she could wait here until he'd passed, or she could try to escape into the ladies' room across the way. If he saw her, though...
Have fun, and keep writing!