Eccentric training takes place when a muscle is under tension while it is lengthening. For instance, the eccentric phase of a bicep curl is when you are lowering the weight (in a controlled manner).
The greatest benefit of eccentric exercise training is the ability to produce the greatest magnitude of force with the least amount of effort. Since we are able to produce much greater force eccentrically, we can stress our muscles much more to stimulate muscle growth, and increase strength gains much more efficiently than concentric exercises (i.e. the lifting phase of a bicep curl).
Eccentric training is great for rehabilitation as strengthening the muscle eccentrically will help prevent injury. When done properly, eccentric exercise is great for older adults as it generates a lot of force with minimal energy. This can help with bone density and muscle mass.