The roots of potted trees are limited by the size of their container as to how far they can spread. After a period of time which varies dependant on the species of tree, the root systems of all pot-grown trees fill their containers, eventually becoming 'pot bound' and in some cases they begin to push the tree out of the pot.
When this happens the fine feeder roots that are essential to the uptake of water and nutrients have little room to grow, are far away from the main trunk of the tree, the soil structure deteriorates and the plant starts to suffer. At this point repotting is required to allow fresh compost/potting medium/substrate to be inserted in and around the root system and the root sytem itself may require cutting back so that it can continue to form fine feeder roots closer to the trunk.
As the container size is part of the overall design, so roots will generally need to be pruned at the time of repotting. Different species of tree react differently to root pruning and a good knowledge of the growth patterns of the tree species helps to guestimate how much of the root mass can be reduced.
As a side note when we do a workshop at Gardening Scotland we are always asked by the public about repotting Acers in patio containers and most people think that they need to buy a larger pot and add fresh peat based compost. Some report that they have purchased huge containers at enormous expense. Our advice makes them at least think about removing the tree from the patio pot combing the old compost out from the root mass, reducing the size of the root mass (by about a third) and repotting the acer in the same pot with a soil mix which is at least 50% grit/cat litter. Bonsai enthusiasts of course would use their own preffered soil mix but then they would also regularly water the tree which is normally not the case with patio plants.
In addition to the very good Graham Potter (Kaizen) guide signposted here, there is also a good Graham Potter video on the Kaizen site and there is also a useful 3 part guide on www.bonsai4me.com