Training pots are typically oversized pots or boxes used for the expressed purpose of obtaining faster growth, trunk and root development than would be unobtainable in a regular bonsai pot.
Although growing in the ground will achieve these results quicker, some people do not have the space to maintain a growing bed.
Training pots are also used to accustom a plant to life in a container.
Common procedures for developing a bonsai from a seedling involve first ground growth until desired trunk size is achieved. Then transfer to a growing box or pot to further define nebari, trunk and create finer "feeding" roots while accustoming the plant to a container. Finally transferring the plant to a bonsai pot where it will spend the remainder of its lifetime.
Any large container will work for this purpose. Some people use a shallow box made from wood, others use oversized azalea pots, and some just use regular clay pots. Dish washing tubs, troughs, and small tires have also been used with some success. Finer root development can be obtained by using a pot with screened sides and bottom. These are available made with redwood or you could also use pond baskets, which are not as attractive but work the same and are available at low prices at most pond stores.
In the first picture you’ll see a screen sided redwood training pot. Vance Wood patents this particular design. Screen sided pots are reputed to help in root development as when the roots reach daylight they ramify. I have had excellent results with screen sided pots. Vance’s Redwood pots are very attractive and expensive. They look great around the yard but discolor rapidly especially if you mulch or bury them in the winter.
The second picture is of a pond basket, sold at most garden centers to be used for plants growing in ponds or for filtering. These come in various sizes and are quite inexpensive; the 2-gallon size shown was purchased for under a dollar. These have all the advantages of the redwood pot except the appearance. They can be buried for winter if needed and like the wooden framed pot, has excellent drainage all around. These are my favorite and are quickly becoming very popular.
The third picture is of basic garden center plastic liners. These do nothing for root development and are not usually recommended. You can group most planters, pots, and containers in this catagory.
The best alternative to growing in the ground is of course the wooden growing box. This subject has been covered quite well
here.
The common denominator is using a container that will allow good drainage, root ramification, and that is reasonably affordable. Not to mention bringing growing conditions as close to actual in ground growing as possible.
EDIT:
Correction as pointed out by Bill Struhar.
Vance's design is intended to combine excellent drainage with air pruning of roots. The root tips that reach the air through the screen stop growing, which allows resources to be redirected to other roots deeper within the soil. Thus, the ramification which occurs due to growing in a screened container does not occur at the screen, but rather in the interior (where we want them).
Will Heath