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
Knowing what weather is associated with each point also seems to be important and this is largely due to whether clouds have formed - the animation covers this quite well. When Tm touches the ground you get nice warm(ish!) weather, with clear skies so no rain. In complete contrast at the cold front where winds are strongest you get cumulonimbus clouds forms which bring the heaviest rainfall. During occlusion as high pressure starts to be created and moisture leaves, you get steady light precipiation and winds. The basic idea is that weather changes as a depression travels over an area and matures!