This is a great question (and a hotly debated one too!). First of all, let’s define cover crop. A cover crop is any crop (or mixture of crops) that are grown to improve the soil. Legumes (bean and peas) are often used for Nitrogen fixation. Grasses (grains like wheat, oats, barley, rye) are grown to prevent erosion and increase the penetration of air into the soil. We always like to throw in some wildflower seed or old vegetable seed just to see what comes up.
Once your cover crop has grown for a few months (usually it is sown in the late Fall and tilled-in in the late Spring), you need to decide what kind of plants you are going to plant after the cover crop. If you are going to grow annual vegetables, you should turn in your cover crop when it is about 50% in flower.
If you wait longer than that, the plant starts to pull Nitrogen back up from its roots for seed formation and takes longer to break down and release into the soil. If you are going to plant perennials (herbs, flowers, berries, fruit trees, etc.) it’s fine to wait to turn the cover crop in for a few more weeks. The more woody and fibrous a plant is, the longer it takes to decompose which offers a slow release of nutrients to adjacent plants.
Other considerations with turning in cover crops include providing food for pollinators and saving seed. It is often very hard to cut down your cover crop when it is covered with bees, butterflies and ladybugs. Also, if you are interested in harvesting some seed from your cover cropped area, then letting it mature or “senesce,” is essential before chopping it down.