The following series of pictures illustrates the practise of root pruning and repotting bonsai. The tree in this series is a Fuchsia magellanica that has been in its current pot for 4-5 years, as a very fast growing tree, annual rootpruning is necessary.
 

Before Removing The Bonsai From Its Current Pot

Before the process of repotting begins, it is always worth preparing the materials that will be needed, as time spent looking for materials during the course of repotting prolongs the amount of time the roots are exposed to the air.

Ensure that the following materials are to hand; sufficient good quality bonsai soil, tools, plastic mesh and wire. If a new pot is to be used for the repotting, prepare the pot.
 

Preparing The Pot

If a new pot is to be used, this procedure can be carried out before the tree is lifted from its old pot. If the old pot is to be retained, it will need to be thoroughly cleaned with water and prepared after the tree has been removed.

These three images show how the pot should be prepared for the tree. Plastic mesh is used to cover the drainage holes of the pot. The mesh stops the soil medium from falling through the drainage holes whilst still allowing the free drainage that is required. 'Butterflies' are shaped from bonsai wire to hold the plastic mesh in position. Lengths of wire or string must also be threaded through the drainage holes in a loop (not shown in this picture) to be used for tying the tree into position so that it cannot become dislodged within the pot. In pots over 2" deep, it is also worth putting a shallow drainage layer of grit at the bottom of the pot at this stage, though in this case, the pot is too shallow to allow room for this.

Preparing New Soil For Repotting

There are a very large number of soil mixes that are suitable for bonsai. Indeed, it is a hotly debated subject within bonsai-circles as choosing the right soil-mix for your tree and climate is of the utmost importance.

It should be noted that it is essential that the soil that is used is free-draining and does not compact easily. Never use garden soils or ordinary potting compost as they are not adequate for bonsai cultivation. For a more detailed examination of Bonsai soil mixes see Bonsai Soil Mixes

Removing The Tree From Its Pot

Cut the tying-in wires from the bottom and tilt the tree out of the pot. If the tree is reluctant to come out, try tapping the sides of the pot with your hand to try to separate the soil from the edges of the pot. If this fails to work, run a sharp knife along the inner edges of the pot to release the rootball. Gently, lift the tree to inspect the rootball.

In this picture of the rootball of the Fuchsia, it is possible to see that there is no room for root growth and the root system has filled the pot and taken its shape in only one growing season.

Removing The Old Soil And Combing Out

After removal of the rootball from the pot, it is now necessary to comb out the rootball. This not only removes much of the old compost but also disentangles any long roots that will need to be trimmed back.

Remove as much old soil from around the edges of the rootball as possible, using either a pointed stick or by hand; chopsticks are useful for this job. Metal roothooks are often used by enthusiasts but in my opinion it is too easy to damage roots this way; roots end up being torn and not cleanly cut.

Particular care must be taken to protect the nebari.

After the old soil mass has been removed and the new root growth has been disentangled, excess circling roots can be trimmed back with a sharp pair of scissors or shears.

The amount of root that should be removed depends on a number of variables including vigour of the tree, the density of the rootball and according to tree species. However as a general guideline, aim to remove around 1/3 of the overall rootmass.

The remaining root system should be carefully examined for any root problems that may exist;

Remove any dead, decayed or injured roots to prevent or cure problems with root rot. Dead or rotted roots will be black, slimy and their outer bark will slip easily from the root itself; severely rotted roots will be entirely hollow and crumble away. All signs of rot must be completely removed to prevent its spread.

Examine the rootball for any other signs of infection or infestation, if found, these should be dealt with appropriately.

After excess circling roots have been removed; check the rootball for faults, particularly around the area of the nebari.

If the rootball has become excessively dense, make wedge shaped cuts into the remaining rootmass. This is only necessary on a well-developed, densely packed rootball and ensures that fresh soil is applied to the centre of the rootmass and that water is able to permeate.

Try to encourage the development of the rootball each time the tree is repotted.

The trunk should have roots spreading radially from around its base; roots that grow upwards or recurve from the base (nebari) are considered ugly.

Strong, thick, downward growing roots should be removed so that the rootball is flat and can be fitted into the pot. Downward growing roots left without pruning will start to lift the tree out of the pot.

Any other thick or straight roots should also be pruned back to a point where there are fine roots branching out. Thick or straight roots tend to rob the vigour of smaller finer growth. At all times when pruning back such roots, it is important that they are inspect carefully to ensure that their removal will not reduce the rootmass to a level where it cannot support the tree.

Thick roots should be cut on a slant with the cut facing downward; this prevents water from accumulating on the cut-surface. Cut the roots cleanly with a sharp knife to help prevent rotting and accelerate healing. Thick cuts can be either sealed with cut-paste or preferably dusted with hormone-rooting powder as it normally also contains a fungicide that will help prevent infection to the root. To accelerate healing and rooting of a thick root that has been cut; it can be worth dressing the cut surface with a thin layer of long-stranded sphagnum moss.

Repotting

After the tree has been rootpruned, it is then necessary to repot the tree. If the pot has not already been prepared, it should be done so now.

Cover the base of the pot (and drainage layer of grit if this is used) with a layer of soil creating a small mound where the tree is to be positioned so that when finally planted, it sits above the height of the rim. Place the tree in the pot and ensure that the correct front of the tree is facing forward. Tie it in firmly with the anchorage wires so that the tree is unable to be rocked about by the wind in the coming weeks whilst new roots are growing.

Add some soil and work it around the root mass carefully so that there are no air pockets. Make sure that the soil is not compressed and take care not to damage the roots. Continue to add soil until the pot is filled just below the rim. When the soil is fully worked in, water the tree thoroughly to ensure that the soil is fully wetted and any remaining air pockets are removed. Watering will settle the soil and it may be necessary to apply more soil to the surface. Re-water until it is certain that the soil has settled fully within the pot.

If using an organic, peat or soil based compost, ensure that the soil is not compressed so much that air and water can not enter the soil-mixture. The tree should be held in position by the anchor wires, not by the soil.

Aftercare

Most trees will show no reaction to repotting and continue on the through Spring without any problems. Some extra care should be taken however in the six weeks after repotting to ensure the health of the tree;

The requirement of water by the tree will be lessened and though the tree should never be allowed to dry out, ensure that it is not overwatered.

Avoid exposure to severe frosts, the tree should be regarded as less hardy than normal for six weeks after repotting.

Do not place the tree where it is exposed to strong winds or sun. This is particularly necessary with evergreens as the increase in loss of moisture through the leaves as a result of the wind and sun will increase the stress on the newly pruned roots. It is possible under windy or hot conditions for evergreens to lose foliage if the reduced rootball is not able to replace evaporated moisture. If foliage does start to dry out on evergreens, provide a shady position out of the wind and mist the foliage regularly.