A good serie of questions that deserve a detailled answer!
The supply level of a given region comes from 2 sources:
a) what the countryside gives. This amount is generally small, and is further diminished by bad weather, pillage, or a military control under 100%
b) what the structures of the region (if you control them) give. The PDF on structures list the precise amount, but suffice to say that high level cities, unblocked harbors and depots are the best.
The amount generated each turn, which is not stockpilable is a quantity in supply chips (SC). A Supply chip contains 5 food&water supplies, and (in addition) 2 ammo supplies, if the region contains a structure (except level 1 town). => You wont get the 5/2 supplies chips from a region without structures.
These supplies chips are then used by units (including supplies units) to resplenish themselves (click on the NATO symbols of the various companies and sub-elements, at the far right of the screen, when you have a unit selected to have the detail window pop-up and show you the supply usage and maximum stock).
Additional notes:
Supplies chips can be drawn from adjacent regions.
If there is a lack of SC, then Supply units will be used to provide supplies (the amount they have is in supply points, like all units, and not in supplies chips. If I recall well each element of a supply unit contains 80 general supplies and 20 ammos supplies. So a full supply wagon, with 4 elements holds 320/80).
Supplies units can shield an army from bad weather hits, and it will use an additional 5 supplies points for each hit prevented.
Hope this helps (and as usual, proof reading is welcome, we are making a detailled game manual in PDF format...)