Hybrid picking is a picking style where you use both, the pick and the remaining(free) fingers, to pick the strings. It is a technique often used in country, but also in other genres. It can give a very different feeling to a sound. It is used to jump from strings who are widely separated. For example a lick that contains notes on the 6th string and on the 2nd string.
In this picture you can see the man picking (with his pick) the 5th string. With his middlefinger he reaches for the 3rd string.
Chicken picking is a subtechnique within hybrid picking. A chicken picker mutes the fingerpicked strings right after he plucks them. Like that you create a dry 'tick' sound. This technique is very commonly used in country and bluesgrass. Another word for chicken picking is Travis picking.
Examples of hybrid picking guitarists
A video demonstration of chicken picking, by James Burton