S.4  SUPPLY LINES
Supply lines are formed on land by a chain of retreat locations, in the opposite direction to how you move when retreating (so that you can follow the chain of retreat locations back from the end of the supply line to the start). The end of a supply line will normally be an army, and the start will be a supply centre (your capital or a naval base). There are no supply lines at sea.
Note: Because the retreat location for an area is always updated when your army moves (it will be wherever the army moved from) your army effectively leaves a trail of retreat locations behind it. Armies retreat back to where they came from - unless you write a order something different. And supply trains move up the trail in the other direction.
For each land area the game report shows where an army in that area can draw supply (tracing a chain back to a supply centre, so that if you build supply in the supply centre shown you can then move it to the army - provided no-one else gets in your way).
Note: An area doesn't need to have the retreat mode set in order to use the retreat location to form a supply line.

S.5  ARMY MOVES
Armies don't need supply trains to move, but supply trains reduce the cost. The normal cost of moving on land is 1 BP per army moved, but this is reduced by 1 BP for each supply train in the area being moved from (but a move action always costs at least 1 BP no matter what).
One supply train in expended in each move, and a move from an area with no supply trains (that can't pay the supply cost) disperses one army instead.
Supply isn't needed and has no effect on the cost of moving at sea, for either armies or ships.

S.6  ATTACKS
The normal cost (in BPs) of attacking on land is 1 BP per army, but this is reduced by 1 BP for each supply train in the area being moved from (except an attack action always costs at least 1 BP no matter what). One supply train is expended in each attack, and attacks from areas with no supply trains are not allowed. 
In a successful attack on land an extra supply train (if available) is expended to move the entire attacking force together (instead of dividing it). Supply trains cannot retreat. Any in an area that's attacked suffer the same collateral damage as population and forts.
Supply isn't needed and has no effect on the cost of attacks at sea, or when attacking from the sea to the land.

S.7  MOVING SUPPLIES WITH ARMIES
In all land moves and attacks any supply trains that aren't used continue to move together with the armies, except that if any armies are left behind then one supply train is left behind as well.
The reason for leaving one behind when you leave troops behind is so that they can still move (without dispersing) or attack. You'll also find it useful if your army has to retreat, since retreating armies always leave their supplies behind.