Clay particles are like flat plates that stack one on top of another, often too closely to let water drain through. You might think that adding sand would help, but often the result is something like concrete. Organic matter, and the humus that it produces when it decays, helps to break up the solidity of clay as well as bind clay particles into small groups that behave like particles of sand. Add as much organic matter as you can, especially to soil that refuses to drain.
Adding Organic Matter
Spread compost, cow manure, peat moss or other organic matter across the top of your soil in a layer 2 to 4 inches thick. The more sticky the soil, the more organic matter will be necessary to help it drain.
Dig a slice of soil topped with organic matter out of the soil and drop it onto the tarp. The slice should be an inch or two wide. With a garden fork, mix the two layers together. Dig another slice, drop onto the tarp and mix. Continue this process across the bed you're improving.
Place another 2 or 3 inches of organic matter into the hole left by your digging. Using the garden fork, mix this into the soil below.
Put the soil on the tarp back into the hole and level it with a rake. Break up any clods of clay and mix them with the loosened soil. You will have a slightly mounded bed of soil with lots of air mixed in. This will settle over the next few weeks, but don't push it down. Preserve the air spaces.