Swales in permaculture are on-contour, or level, ditches. Imagine a line drawn across a hilly piece of land that is perfectly horizontal and level. Then, imagine a ditch there. That is an on-contour swale.
When done right, water hitting that contour from anywhere on the slope will spread out evenly throughout the swale and hydrate the entire hillside. These swales do a wonderful job of slowing, spreading and sinking water into the landscape.

When done right, with the right soils, swales can hydrate not just the swale area, but a whole landscape because water tends to form a lens under the earth, headed down hill.

In stormwater management circles, swales are considered to be ditches somewhat off contour. They still slow and spread water but instead of spreading across the entire landscape, it drains to a single point. These are used to prevent erosion and control the flow of water. They are sometimes planted with wetlands plants, as in the above photo.

In Florida, drainage swales are common and are healthier than ditches that lead straight to storm drains. There is a chance for some of the water to infiltrate and get filtered and cleaned out by the wetlands plants and the soil. You may find a drainage swale useful if you have areas that flood that you’d like to keep clear. A permaculturist will look at how that can be done in the most regenerative way possible.
Swales are a very popular idea in permaculture, but they don’t belong everywhere and aren’t needed everywhere. They are a site specific solution.
The concept can be used even on flat land. For instance, I used my garden paths as swales. I caught water from my roof, stored it in tanks, and “flood irrigated” my garden in the dry season from the tanks.
My paths were on contour, so the water sent into one path would spread throughout my garden. The ends of the pathways were bermed so the water would stop there, and soak into the sides of the raised beds.
I put banana leaves and matted grass on the paths to slow the water from soaking into the path and encourage it to soak into the beds. It worked pretty well and watering was fast and simple. Not the most efficient way to water (drip irrigation would have been a more frugal use) but we had lots of water storage and we needed no irrigation equipment to set it up.
Hands On Activity
Most places have evacuation zone categories, especially in hurricane country or flood prone areas. Do a search of your county name + “evacuation zone” and you should find a map or database that you can enter your address into. Note what category of storm or hurricane would cause you to need to evacuate to be safe from storm surge (if any).
Do a search in FEMA’s flood map database or floodfactor.com for your site. Enter your address or find the site on the map. Note if you are in a flood zone or not, and what type it is. Note: these flood maps have proven to be inadequate for some areas that have had recent major floods and new ones are being created but these give you some idea of where water ends up.
Observe and record where water flows and where it collects on your land. Think about how you can use that energy in your design. How can you place elements where they can benefit from water? How can you direct water to where it is needed most? How and where can you store it?
Don’t get too locked into any one solution or plan yet. Try to think of multiple options of how and where you could move water at this point.