The Five Basic Bonsai Styles: Formal Upright, Informal Upright, Slanting (or windswept), Semi-Cascade and Cascade.

What ever the style and its requirements, it is important to remember that in all types of bonsai, neither the trunk or any of the branches should point towards the viewer when the bonsai is viewed from the front. 

Formal Upright Bonsai:  Goal is to simulate what occurs when a tree has grown in the open under perfect conditions.
The key requirement for this style is that the trunk be perfectly straight; about one third of the trunk should be visible from the front. 
The trunk must taper in a naturally look evenly from base to apex. The Taper of the trunk becomes very distinctive because it is achieved by cutting off the growing tip of the trunk or branch with each new year and wiring a new branch into position to form the apex. This is something quite hard to do, however it produces a stunning result when the trunk starts to mature and the taper starts becoming prominent.
The branches should be symmetrical spaced; they can either be positioned directly across the trunk from one another or alternating, each branch being higher along the trunk then the branch below it on the opposite side.  The first branch up from the bottom is the longest and in proportion and usually at a near right angle to the trunk.  This first branch is usually trained to grow to an equivalent to a third of the total height of the tree and is the largest branch.  Over all, the branches do not need to be symmetrical in length as long as the over all look of the Tree is balanced when viewed from any direction.
The Crown or top of the bonsai is usually very thick with foliage - so full and tightly ramified that it is difficult to see its internal structure through the mass of leaves or needles. The tip of this style of bonsai also has a slight curve, to lean forward usually towards the viewer.
Recommended Species: Pines, Junipers, Spruces, and Larches; almost any species that has the flexibility to be trained into this king of ridged style. Conifers do best, but Maples can also be used, however maples are not as easy to train.  Most fruiting or naturally informal trees are not suitable for formal upright styling.

Informal Upright Bonsai: Goal in this style is to simulate a tree in nature that has had less then ideal conditions and has had to bend and later direction due to wind or shoade or other obstructions in order to get adequate light. 
The key requirement is that the trunk must bend slightly to the right or left - but never towards the viewer. In nature, such trees bend or alter their direction away from wind or shade other trees or buildings, or towards light.  An informal upright bonsai basically uses the same principles of the formal upright bonsai only that it is informal.
The trunk  is still required to be tapered, however the trunk direction and branch positioning is not straight  The trunk bends and twists from side to side in a series of curves, in what is considered an informal manor;  closer to the way a tree would look when exposed to the elements at an early age.
 The branches are positioned along the trunk to balance the bending and twisting effect of the trunk.
The crown or top of the tree is mainly very full with foliage and despite the informal trunk, is most always located directly above the base of the tree. This is an attribute of the informal upright style;  it needs to be done this way to prevent the look of slanting
Jin; the carved remains of dead or unwanted branches, made to look like dead and rotting limbs, also more appropriate and effective with the informal upright style

Recommended Species:, Japanese Maple (Acer palmatum), Trident Maple (Acer buergerianum), Beech, practically all Conifers and other ornamental trees such as the Crab Apple, Cotoneaster and Pomegranate.  In general most species are suitable for this style.

Slanting Style:  Goal here is to simulate a natural tree that slants as a result of wind or deep shade during its natureal early development.   The key requirement is that the trunk must lean at an angle up to a maximum of 45 degree of vertical away from the root base. This style has similarities to the informal upright style in the trunk, branch and crown treatments.  This style is a fairly simple one that can be achieved by a number of methods.  On method is to wire the trunk at an early age until it is positioned as you like.  Another is to put the actual pot on a slant, causing the tree to grow normally upward to the sun, but abnormally in relationship to the pot. Changes in the placement of the pot angle can curve and twist the trunk.
The trunk can be curved or straight, but must be on an angle to either the right or left, never to the front, with the apex not directly over the base of the bonsai.  The whole trunk leans at a definite angle from the base to the just below the crown.
The branches can be directly opposed or alternating
The stronger roots grow out on the side, away from the angle of the trunk lean, to support the weight.
The crown is also full and can twist back towards the base if desired.
Recommended Species: Most species are suitable for this style, as the style does bear similarity to informal upright. Conifers work particularly well.

Other Notes on formal upright, informal upright and slanted styles: the number three is significant. The lowest branches are grouped in threes, and this grouping begins one-third of the way up the trunk. The bottom-most three branches almost encircle the trunk, with two branches thrusting forward, one slightly higher than the other. The third branch, emanating from a point between the first two, is set at such an angle as to make the foliage appear lower than the other two. This pattern presents an easy way to tell front from back and sets the tone of the entire composition.

Cascade Style:  The Goal of Cascade style is to simulate a tree growing from the side of a Mountain or Cliff.  The trunk has a natural taper and gives the impression of being pulled down and twisted by the forces of gravity.
The key requirement of this style is a tall, narrow pot which will enhance the style and accommodate the cascade and a species of plant that will adopt this style if trained.  The main trunk should be wired to spill over and down the edge of the pot; the main focus is on the first major bend which should form an upside-down U shape.
The trunk line of this style curves downward and the trunk ends below the Base of the pot or container.  The main trunk is winding like a rock fall or a stream meandering down the side of a mountain. 

The branches are sparse, smallish for the length of the trunk and elegant; appearing to be seeking the light twisting upward and outward from the trunk, reaching. Emphasis should also be kept on keeping the branches uniform and horizontal to the almost directly vertical trunk.  Branches are sparse; the trunk is a main focus of this style bonsai.

Recommended Species: Many species are suitable, if they are not strongly upright.
Remember is that both cascade and semi-cascade should be positioned right into the center of the pot, the opposite to what you would do for any other style.

Semi-Cascade:  The goal of Semi-Cascade is to simulate the trees growing along a stream or natural water side; trees that reach out from the bank away from the shade, towards the stronger sun over the water.
The key requirement is that the trunk appears almost horizontal and the plant can grow a little below the level of the pot rim. 
The Trunk tip of a semi-cascade, like the cascade, projects over the rim of the container, but in the semi-cascade does not usually drop below its pot rim.  The angle of the trunk in this bonsai is not precise; as long as the effect is strongly horizontal.  Any exposed roots should balance the trunk.
The Branches as with the branches of the cascade tend to be small and short.  They alternate around the trunk, but do not face front directly. 
Recommended Species: Many species are suitable, except strongly upright ones. Flowering cherries, cedars and junipers work well.

Brief Mentions of Other Bonsai Styles

Broom Style:   This has been referred to as a restrained classic style.  Named after the shape of an upturned Japanese broom; this style consists of an straight upright trunk topped by a domed head of fine twiggy branches.  It is a difficult style to achieve and works best with deciduous species with fine branches such as elm trees.

Literati Style:  Also called Bunjin  This style takes its inspiration from Chinese Painting of Scholars called Wenjen; the style simulate trees grown along seashore or where they need to reach for strong changing light.  The trunk line flows and twists in several sharp’ish curves; the goal is a refined elegance.  Species of conifers and deciduous tresses such as Hawthorn work in this style.

Root Over Rock:   This style simulates in nature what happens when trees grow in a crevice or from a rocky ledge sending out roots in search of moisture and nutrients in deeper soil.  The main design feature of this style is the roots clinging to the rocks.  Works well with trees that have strong roots and grow easily on rock such as Chinese Elm , Junipers and Pines . 

Clasped-to-Rock Style: this style simulates trees growing on mountains and cliffs.  The roots in this style are confined only to the rock and do not extend into the soil. Plant your trees directly on a slab of slate or on a rugged piece of rock; display in a shallow tray of sand or water.  Recommended species are native to mountainous areas, such as juniper, pine, birch and spruce.

Twin Trunk Style:  Two trunks growing from the same root system.  One will be the dominant, in a upright or semi upright style with the other in a version of the slant style; though divided trunks can actually be done in any style.  Birch, Beach, Cedar, and Maple all make good choices for this style.

Clump Style:  characterized by several trunks growing from the sam root in a clump formation.  Accrues in nature when a main tree is cut, lighting blasted or other wise has its original main trunk destroyed. 

Straight Line Style:  Also call Raft Style; this style is based on a natural tree that has fallen or blown over yet continues to grow.  The original branches on the up side reach skyward vertically to become new trunks.

Sinuous Style:  This style occurs in nature usually on plants that have shallow root systems that will produce suckers and or plants which will root low hanging branches to form new trunks.  Flowering Quince and Azalea are two good examples.

Saikei Planting:  Also called a “tray landscape” The goal is to present a natural landscape in miniature.  Sand, pebbles, rocks, mosses, plants of various sizes can be used.  Often short term compositions, the plants can be transferred to individual bonsai pots when they become too mature.

Group Planting:   The goal is to create several trees growing together such as a birch clump or to create a miniature forest-scape.  The key is to make the arrangement look natural and uncontrived.  Using Odd an number of plants is recommended; most species can be planted as groups.