First of all, on the board index, use ctrl-f to search for the term "dry well" (Might be listed as "drywell")
What you are looking for is a pit with enough room to hold the rainwater while it drains into the soil. It would be easier to dig a pit like this using a backhoe, which is why I suggest renting a bobcat. (besides, they are a LOT of FUN to play with!)
You are right to look for a solution, because your runoff is typically your liability and in many cases your neighbor could go after you for damages.
The size of your drywell will depend on how big your roof is, how much rain you get, and how fast the water drains into the soil. Your local building department should be able to help you with the answer, as well as provide you with requirements for setbacks, burial depth, size, etc. You may find that you need a permit to do this.