Most towns/cities will not be too happy with you, if you start cutting up their road. So you'll have to use a method that will be completely on your property.
Dump some dirt
The easiest solution would be to build up the area with a load of dirt. Using a wheelbarrow and a shovel, grab some dirt from another location in your yard. If you can't find a place in your yard to take dirt from, you could always purchase some from a gardening/landscape/home improvement store. Dump the dirt along the edge of the road until your hole becomes a hill, then feather it into the rest of the yard. Plant some grass, and grab one of @dbracey's margaritas and watch it grow.
A french drain would give the water a place to go.
Start by digging a trench right next to the road about 12" wide, and 12" deep (don't forget to call before you dig, so you don't damage any underground utilities). You want the water to drain away, so you'll want a drop of at least 1/8" for every 1' in length. For example. If the drain will be 20' long, the end of the drain will be 2 1/2" lower than the beginning. Next fill the trench with about 4" of crushed stone, and lay your drain pipe in the middle of the trench. You can use either rigid, or flexible pipe.
Fill the trench to about 2" from the top with more crushed stone, then finish filling the trench with topsoil. plant grass, and enjoy.
If you're familiar with working with concrete, and you like a more finished look. You could install a gutter, along the side of the road.
Just remember, you'll need the gutter to slope so water will run away. It's probably a good idea to install a dry well or other drainage method at the end of the gutter, so the water has a place to go (unless you don't care if you simply move the puddle to your neighbors yard).