A great example of this in a biological system is with oxytocin and childbirth. Oxytocin is a hormone that is secreted from the pituitary gland and causes contractions. When the baby pushes against the cervix, nerve impulses tell the brain to release more oxytocin, which causes further contractions, and so on.

In nature, the melting of sea ice is an excellent example. When ice warms up, it can melt. The water has a lower albedo (is darker) than the ice, meaning is absorbs more solar radiation and gets warmer. This results in more melting, which lowers the albedo further, and results in more warming...

A simple example of negative feedback is body temperature. If you get too hot, you start to sweat in order to lower your body temp. If you get too cold, you'll shiver to bring the temp up. 

The classic example of this is with blood sugar. The pancreas serves as the receptor - the organ that detects/receives the stimulus. When our body notices that blood sugar is too high, the beta cells in the pancreas work as an effector - the thing that does the response. In this case, the response is to lower the blood sugar by releasing insulin and causing the glucose to be stored in our liver and skeletal muscles as glycogen. If the blood sugar drops too much, the pancreas's alpha cells will release glucagon, which causes the breakdown of glucagon into glucose, which is then released into the blood.

In nature, a great example of negative feedback is with cloud formation. When temperatures rise, evaporation increases, which results in more clouds being formed. With more clouds, more solar radiation gets reflected back into space rather than making it to Earth's surface, resulting in lower temperatures and less evaporation.