According to the Missouri Botanical Garden, rainscaping is "any combination of plantings, water features, catch basins, permeable pavement and other activities that manage stormwater as close as possible to where it falls, rather than moving it someplace else. In addition to rain gardens and bioswales, a diverse landscape that includes trees, shrubs, perennials, mulch and amended soils intercepts and disperses rain as it falls, and allows more water absorption into the soil and by plants."
A rain garden collects water from flat surfaces (e.g., roof or driveway run off) and redirects it into a shallow depression. This allows the water to slowly soak into the soil. For general advice about rain gardens start here, here, and here.
Deep rooting, fibrous plants are good choices for rain gardens and include trees, shrubs, perennials, and annuals. Rocks and various soil amendments are also helpful in creating this stunning and functional addition to your landscape.