Subsurface drainage is drainage that occurs under the ground. The water is collected through various ways such as gutters, catch basins or perforated tubing and then send the water to the desired location. The is the most common for home owner because it keeps the aesthetics of your yard without digging ugly ditches like you see on the side of the road. The image above shows a french drain system where the water is absorbed through the ground and then into the black drainage pipe that has little holes in it to collect the water. The sock or black mesh material around the black corrugated pipe is to prevent mud and other debris from clogging the small holes in the pipe that collects the water. The drainage rock that is around the pipe helps fast drainage as well as a barrier to keep mud and other debris from clogging the pipe as well.
Surface drainage is all type of drainage that's visible. Ditches are a prime example of this. You can see the water flowing through the ditch or culverts when it rains. This isn't the only type though. Channel drains are also apart of this category. Channel drains are often used around pools to keep water from going towards the home owners house. The image above is a channel drain put into concrete, but they can also be put directly into the soil/ground. They are great for large amount of water drainage since the entire length of the drain is open. Another example of this being used would be to put a channel drain in front of your house or garage to keep water from flowing into it. You would put the channel drain the entire length of the garage to stop the water.