Erica's Original Notes 
This year's leaf buds are the result of last summer's work; the precursors for next year’s foliage are protected in a variety of tough, waterproof materials. The dormant bud lives in miniature without stirring for up to seven months, at a standstill, with minimal food requirements. In winter, the branches are studded with life (axillary or lateral buds) that are arranged very precisely and borne just above last year's leaf scar.

Buds may be:
a) axillary - In spring not all buds open; some stay dormant until needed, e.g. if there is loss of a branch, etc. They may ultimately shrivel and die;
b) terminal - the end bud that grows into the mainline and is therefore larger;
c) flower buds - remain closed until the ovary is old enough to consider matrimony!

Variety of Buds
Aesculus has moist 'cotton batting' (hairs) which keep it from drying out.
Caragana has shaggy, untidy buds.
Catalpa: no terminal bud; instead there are axillary buds in whorls of three, minutely borne above leaf scars; fun to watch develop into those enormous leaves!
Cornus: The covering bud scales are softened by sap and separate neatly, growing like a fingernail and seen as discoloration at the tip of the sepal.
Fagus: The buds are alternate, pointed and divergent (stick out) with 10 to 12 overlapping neat scales.
Fraxinus have chocolate chip-like buds, due to flattened protective hairs.
Ginkgo: stubby, like Larix.
Larix has knob-like buds, with two sorts, one producing tufts of needles and the other slender branchlets with individual needles.
Liriodendron has two scales `in prayer'.
Magnolia: The leaf scar encircles lateral buds which usually have two furry scales.
Prunus: The buds are typically clustered at twig ends.
Quercus: Buds also cluster at tip of branch.
Salix has only one bud scale.

Updated 2013 (RP)
Dormancy of buds is controlled by plant hormones known as auxins. The terminal bud is dominant, as this can be important for the success of a plant if it needs to grow upwards and away from competition for light with surrounding plants. The terminal bud produces hormones that inhibit the growth of the axillary or lateral buds (found in the angle between the leaf and the stem). Thus if the terminal shoot of a plant is pruned, the source of the inhibiting hormone is removed and the lateral buds start to grow.

Updated 2018 
Comparison of two types of mixed buds: the large, scaly brown buds of  Aesculus glabra, or Ohio buckeye, and the long, smooth light brown (and looking orange against the blue sky) buds of Fagus grandifolia, or American beech.