How do I stop my driveway from flooding?
You have two ways to get rid of it.

Pump the water away.

This will allow you to move the water to a higher elevation where it can drain away naturally. See sump pumps.

Create a drain for the water.

You'll need to determine the best place for water to go. This depends on what is even further from your house. It looks like in the pictures that the driveway goes slightly uphill away from where the water is pooling and towards the road. If that is the case, trying to get the water to go that way will be hard.
Seemingly more likely, you could have it drain through the back yard. In order to do that, you'll either need to grade the dirt, or add another path for water to flow.
Grading the dirt:
Since you have a low spot next to the drive way, you'll need to pull some from where you intend water to flow into the lower area. Water will flow down hill, and you'll need to make that hill for it to flow down, even if the hill is barely noticeable.
Adding another path:
In this case, you'll dig a trench from the lowest point next to the drive way to either the edge of your property or a point where it will run off easily. You can use field tile (corrugated pipe) to run it in. If you use perforated pipe, you can put some rock on top and around it to let water in. You'll need a path for the water that's sitting to get into the pipe. Also, you'll want to run any downspouts in the area into that pipe to have them drain off too.
Here is a description of how to install corrugated pipe.