A drywell is basically just a large hole in the ground filled with washed crushed stone into which rainwater drains to perk into the ground, and eventually into the groundwater.
Now, that having been said, these things only work where the makeup of the surrounding soil is rather gravelly / sandy and free-draining. That means that if you dig a hole and fill it with water, the water will disappear into the ground and not just sit there. A consideration here is the prec rate of the soil, which has to do with how fast the water disappears, and how much water the soil can absorb before it becomes saturated.
If your soil is more sandy than clayey, then you might have a good chance of making a drywell work. But it would be best to test before going to all the effort. The test can be accomplished right where you might build a drywell. Once you kow if the soil will accept the drainage, then a dry well can be constructed by, as I said above, digging a large and deep (about 4-5 feet) hole. Dump about 2 feet of crushed stone into the hole, and then install a PVC pipe large enough to fit your downspout into, from the stone, back to the house, and up to the downspout. Then dump another foot or so of stone over the pipe, and back fill the hole with native earth. Connect your downspout into the PVC pipe, and you have your drywell. In most situations, wrapping the crushed stone in a filter fabric will prevent the fine particles in the native soil from migrating into the stone and filling the voids. When that happens, it reduces the functional life of the drywell.