The underlying principle of Bonsai, today, is that a bonsai tree is 'styled' to represents the landscape that it may be found in nature.'
A culster of Bonsai Trees grown in a tray or on a rock slab, represent a Forest. Bonsai Grown on a Rockscape; depict trees growing on hill sides or mountain slopes. These representations are called Bonsai Styles.
Formal Upright Style (Chokkan)
The tree trunk is up-right and straight, the trunk has a visible taper with the widest part of the trunk at the based of the tree and supported by a suitable radial nebari (or root structure). In nature this style is found commonly in certain species of trees like some Conifers, and when a tree is growing in a open space and not competing with near-by trees for light, it usually also well protected from the harsher elements. A formal upright tree presents a picture of grandeur, regal maturity and a timeless quality.
Informal Upright Style (Moyogi)
The Informal grows upright, but unlike the formal upright has gentle curves in its truck. The trunk rises out of the soil at an angle and usually curves gently back and forth two or three times before it reaches the apex. This is the most common style and a good style to start ones bonsai beginner to their first bonsai with. Informal Upright Style is suitable for most species of Trees, and is especially suited to deciduous trees and flowering trees like, maples and elms, prunus and quince.
Slanting Style (Shakan)
The Trunk of the Slating Style Bonsai at an angle at and angle to the surface, as though the tree was blow sideways by a strong gust of wind, or knocked over in landslide or by another larger tree falling onto it and knocing it sideways, or some other similar natural phenomenon.
Cascade Style (Kengai)
The Cascade style is easily recognized as the trunk dips below the bottom of the pot. This style represents a tree that is growing on a cliff top and has that cascade over and below the precipice of the cliff top. This style is suitable for most species, but is particularly prevalent amongst juniper, yew, pine and other confer bonsai.
Semi-cascade Style (Han Kengai)
The trunk descends below the rim of the pot but not below the bottom of the pot. Representing a tree growing over riverbanks from a rock-outcrop, where the branches grow almost horizontally over the water surface. This style is suitable for most species, but is particularly prevalent amongst juniper, yew, pine and other confer bonsai.
Broom Style (Hokidachi)
Trees in this style have upright trunk with branches and foliage radiating outwards in the shape of a Japanese fan or an Umbrella.
Literati Style (Bunjingi)
This style is one of the most difficult to describe and is generally considered an Artist style. It is, however, easily recognisable as 70%-80% of the tree is without branches, and the trunks twist and curves several times is usually without taper, and the top usually only has a sparce collection of downwards leaning branches, it pot also is usually very small and shallow when compared to length and height of the tree.
Root over Rock Style (Sekijoju)
The tree is grows over a rock with the roots reaching down to the earth below in search of water. Much like a tree would do if it were grew from seed on top of a rocky out crop. A variation of this style is "clinging-to-a-rock style" (Ishitsuki) where the tree clings to the rock surface, rather than 'sits' on it. Fig trees commonly cling to rocks and other trees.
Root in Rock (Shitsuki)
Tree grown in cracks and crevices in a rock, but the roots are not visible, as they have grown into the rock outcrop rather than over as in the style Sekijoju style
Windswept Style (Fukinagashi)
The trees in this style lean to one side with all the branches on the side to which this tree is slanting. In Nature you find tree like this on gusty hill sides and sea shore, where strong wind damage young shoots on the windward side, but shoots on the leeward manage to survive and grow. This causes the one-sided branch growth and the tree to lean on to one side.
Split Trunk Style (Sabamiki)
Usually has the base of its trunk Split or Hollowed out. In nature this style is usually found in very old trees. In Bonsai Sabamiki can add a feeling of great age to the tree. Sabamiki can be used on both deciduous trees and conifers.
Driftwood Style (Saramiki)
A style of tree where the trunk is mostly deadwood. This style is mostly seen on junipers and yews where the trunk is primarily dead wood with one or two live veins running up to the remaining branches. When accompanied with Jins and Shari this style presents a powerful and dramatic picture.
Weeping Branch Style (Shidare-Zukuri)
The Weeping Branches or Hanging Branches style, like found naturally with most weeping willows, and other pendulous species. The main trunk in the hanging branches styles normally takes one of these four styles upright, informal-upright, slanting or semi-cascade style.
Root in Rock (Shitsuki)
Planted on Rock tree growns in cracks and crevices in a rock, but the roots are not visible, as they have grown into the rock outcrop rather than over as in the style Sekijoju style.
Exposed Root Style (Neagari)
The roots in this style are significantly exposed, much like a mangrove trees. The top portion of tree, above the exposed root, can take any of the any of the more upright or slanting or cascade styles. In nature these trees are often found in mangrove areas, or flood planes or and often on hill sides by a stream where flash foods have eroded away the banks and lets trees roots exposed to the elements, in time these harden and become like extensions of the tree trunk.
Twin Trunk Style (Soju Style)
Is a Tree with two trunks from a common root ball. The two trunks split just above the soil level, and one is usually subservient to the other, with both the trunks contributing branches to in a unified tree crown. Two Trunks Style is usually used in the Wedded configuration with one trunk subservient to the other. An adjunct of Two Trunks is Sokan Style, or double trunk, where one trunk splits just above the soil line and the two branches are treated as separate trees.
Triple Trunk Style (Sankan)
This style may have three, five, or seven trunks of different trunk diameters growing from a single root ball. If there are three trunks of differing diameters, then it is referred to as a father, mother, son arrangement. Slender trunk trees, such as maples and elms are best suited for this style
Multiple Trunk (Kabudachi)
Or Clump Style as it is sometimes called, the trunks seem to sprout out quite randomly from the root base or clump. Often found in nature with Quinces, Spirea and other shrubs or shrub like trees that has been constantly cut down as in hedges.
Raft Style (Korabuki Style)
Raft Style imitates trees in nature that have fallen down and the branches on one side have continued to grow as individual trees. Commonly this style is found in a Straight-line or Ikadabuki imitating a tree trunk, occasionally you see a more Sinuous Root Connected Style (Netsunagari) which is usually imitates a connected surface root that is meandering around in search for and ideal path and which along the way send out new branches that eventually grow into individual trees.
Forest Style (Yose)
Forest Group plantings imitating small clumps of trees groups where each one of them grows as independent trees and competing with each other for its own light and space. Traditionally forest groups have odd number trees, as they have an aesthetic appeal and allow for easy symmetry along and around a focal tree, usually the largest tree in the group. There are two styles of planting Forest Groups, the first, Yoma-Yose Style, where the trees are spread out fairly evenly around the pot or slab, and the second, Tsukami-Yose style Yose, where the trees usually of the same thickness grow outward from a central area, as though imitating trees on a small hill top. Forests or Yose are also often described by the number of Trees they have in the Forest Group, Sambon-Yose (3 trees), Gohon-Yose (5 trees), Nanahon-Yose (7 trees) and Kyuhon-Yose (9 trees).
Landscape Style (Bonkei)
Landscape Style often grown in large shallow unglazed pots and include Rock and Water depicting Mountains and the Sea and are often planted with well placed Bonsai, mosses, grass and other perennials, to depict a more realistic scene.
The Twisted Style (Nejikan)
The Twisted Style is probably the closest to what came over from China; often called the Dragon Style it is a favorite in Chinese Bonsai. A Dragon Bonsai (or Green Dragon) is an auspicious presence in the garden, the mystical Chinese dragon symbolises good fortune and prosperity. In this style the tree coils around itself like a Chinese dragon. The Octopus Style (Takozukuri) a variation of this style often as low sweeping branches, much like the tentacles of an Octopus.
The Coiled Style (Bankan)
Similar to the preceding style with the prime difference being that the bark appears to twist and wind it way up the trunk, much like a snake coiling its way up a branch.
The Style listed above are a few of the Traditional Bonsai Styles, thoungh many more evolving and being widley accepted now and being recognised as modern styles.
My introduction to Bonsai Styles, in the 80's, was throught Peter Chan's Book, Bonsai The Art of Growing & Keeping Miniature Trees, 1985, in which he talks about 21 core styles and all bonsia had to fit into those styles, views, have now changed, and styles have become more indicative rather than the rule.