Yes, it is possible to simulate quantum computations on a classical computer. But the cost of simulations grows exponentially with qubit count and/or circuit depth and/or particular operation counts.
For trying ideas quickly, the simulator Quirk is great. It's an open-source drag-and-drop quantum circuit simulator that runs in your web browser. You can access a live version at algassert.com/quirk.
Here is a screenshot of Quirk's example Grover circuit, which is instrumented with intermediate state displays to track the "hidden" state becoming more likely:
