Eccentric part of a lift, is the downward, or lowering phase of a lift.
For example, with a bench press lift, the eccentric phase is the phase when you lower the bar back to your chest.
Pushing the bar up, off your chest, is known as the concentric lift or phase.

You can train “eccentricly” by having a partner help you lift the weight off your chest, and then let go and let you lower it down on your own.
You can also do it on a machine – put half your normal weight on the stack, push it out with 2 arms, and lower it back with one arm.

It’s the opposite way around with pulling exercises.  Lat Pull Down – the eccentric contraction occurs when you extend your arms and return the weight back to the starting position.

CAUTION – high risk of injury, make sure you are warm and start with a very low weight.

http://bjsm.bmj.com/content/early/2008/11/03/bjsm.2008.051417.short
This study suggests that eccentric training only is superior to concentric training only in terms of increasing muscle mass.  Possibly because you can, in theory, handle heavier weight loads when doing eccentric only lifting.  Again, this increases risk of injury, so be careful.

The aim of this systematic review was to determine if eccentric exercise is superior to concentric exercise in stimulating gains in muscle strength and mass.
Meta-analyses were performed for comparisons between eccentric and concentric training as means to improve muscle strength and mass. In order to determine the importance of different parameters of training, sub-group analyses of intensity of exercise, velocity of movement and mode of contraction were also performed. Twenty randomized controlled trials studies met the inclusion criteria. Meta-analyses showed that when eccentric exercise was performed at higher intensities compared to concentric training, total strength and eccentric strength increased more significantly. However, compared to concentric training, strength gains after eccentric training appeared more specific in terms of velocity and mode of contraction. Eccentric training performed at high intensities showed to be more effective in promoting increases in muscle mass measured as muscle girth. In addition, eccentric training also showed a trend towards increased muscle cross sectional area measured with magnetic resonance imaging or computerized tomography. Sub-group analyses suggest that the superiority of eccentric training to increase muscle strength and mass appears to be related to the higher loads developed during eccentric contractions. The specialized neural pattern of eccentric actions possibly explains the high specificity of strength gains after eccentric training. Further research is required to investigate the underlying mechanisms of this specificity and its functional significance in terms of transferability of strength gains to more complex human movements.

http://onlinelibrary.wiley.com/doi/10.1111/j.1748-1716.1990.tb08973.x/abstract

This study suggests that increases in peak torque and strength-related performance parameters were greater following a programme consisting of maximum concentric and eccentric muscle actions than resistance training using concentric muscle actions only. Because increases in muscle fibre areas were small it is also suggested that the increased muscle strength shown subsequent to short-term accommodated resistance training is mainly due to neural adaptation.

http://jap.physiology.org/content/81/5/2173.short

Strength gains, according to this study are specific to the movement type.
Eccentric training, develops eccentric strength.  Something vital for grappling and MMA.

We compared the effects of concentric (Con) and eccentric (Ecc) isokinetic training on quadriceps muscle strength, cross-sectional area, and neural activation…We conclude that Ecc is more effective than Con isokinetic training for developing strength in Ecc isokinetic muscle actions and that Con is more effective than Ecc isokinetic training for developing strength in Con isokinetic muscle actions. Gains in strength consequent to Con and Ecc training are highly dependent on the muscle action used for training and testing. Muscle hypertrophy and neural adaptations contribute to strength increases consequent to both Con and Ecc training.

The research seems to suggest that training with emphasis on the eccentric portion of the lift will increase muscle mass, and specific eccentric strength.