Depressions form when two air masses with differing characteristics meet. Above the UK this is usually polar maritime (PM) air and tropical maritime (TM) air, which is seperated by the polar front.
Warmer, lighter TM air rises, because, being warmer it has a lower atmospheric pressure so is less dense. As it is warmer, it also carries more moisture than the cooler polar air mass and, as it rises, it cools and condenses and produces water droplets and consequently clouds. This is the embryonic stage and the depression moves in a north easterly direction, under the influence of the Polar Jet, whilst Coriolis establishes the anti-clockwise circulation that permits more Tm air to protrude into the depression. As the depression matures, warm air rises over colver air to the east so lower pressure is created, with adiabatic cooling of air. This adiabatic cooling forms high altititude clouds, followed by lower cirrostratus, altostratus and nimbostratus from which heavy precipitation for a long period of time occurs. Where Tm air remains in contact with the ground, indicating no forced ascent, skies remian clear and cloudless but the faster moving cold front, where strongest winds are generated, force cumulonimbus clouds that prodcude heavy precipitation. Once all the Tm air has been 'forced' off the gournd and the two fronts have caught-up with each other, occlusion occurs. This is marked by a decrease in cloud cover and wind speeds, after a period of steady light precipitation, and an increase in pressue which signifies the decay of a depression.
This is quite a good animation if you are a more visual learner. I think explaining depressions is easiest with diagrams like we used in class but I am struggling to find good diagrams
http://www.ngfl-cymru.org.uk/vtc/depression_and_fronts/eng/Introduction/mainsession.htm

We also have to be able to find the three distinct stages on a synoptic chart - have a go with this one!
The very unscientific way I remember it is that:
Occluded = icicles and suns are fully mixed
Mature = icicles and suns may mix at top but are largely separated
Embryonic = effectively a line of suns and a line of icicles
This is not an easy topic, but hopefully this helps - lets just hope that it does not feature too much in the exam!!!