These go together as a topic since they are generally performed at the same time.
    Root-pruning is performed to reduce the root system in order to re-pot the tree in either the same-sized or smaller pot.  It needs to be done at the proper time, depending on the type of tree, to ensure the tree's survival.

    A little fact about roots: only the growing tips of fine root hairs actually perform the intake functions of the root system... the thick tap root, and the several feet long coiling roots in a nursery pot do very little for the tree.  Helping the tree to replace these long thick roots with a tighter mass of fine roots is just as healthy --- or healthier --- for the tree as it's "natural" root system.

    Do not try to use a schedule for repotting as you may have seen in books; schedules can be used only as a general guidance.  Seasons vary from area to area, and year to year.  When the tree is ready, that is the time (more on this below).
    It is usually required every one to three years.  Some trees are slow-growing and an older specimen would not need repotting as often as a young, fast-growing species.  You have to examine to see if a tree needs root-pruning and repotting.  Roots growing out of the drainage holes is sometimes a sign; loss of vigour, more frequent watering needed, or just having a tight, matted soil in the pot would be signs.  For starting out, just assume once a year, and that is at least a fairly safe way to go.
    Emergency repotting is sometimes necessary due to degrading soil, root rot, soil contamination, or a missed repotting in spring; if it is not the best time to repot, the tree should be repotted with minimal disturbance to the root system.  Wait until a better time to disturb and prune the roots.

    There are two very different root-pruning goals: firstly, the creation of a bonsai from a larger nursery stock or collected tree, and secondly, the maintaining of a bonsai after it's  already been root-reduced and grown in a small bonsai pot.

    Initial creating of a bonsai from a larger tree will require much long, thick root removal and the growing of fine roots closer to the trunk.  Sometimes this can be done completely at one root-pruning, but more often it should be done in stages.  Severe root-chopping can usually be done if the top is being severely reduced at the same time, in the course of initial shaping into bonsai beginnings. The safest way is to cut root masses back in stages, removing the largest of unwanted roots each time.  Repot into "normal" sized pots with sandy soil, encouraging fine root growth, then repeat the process each year.  Eventually, you get the small rootball with fine roots which you can pot into a shallow bonsai pot.
    For a bonsai, you should eventually remove ANY thick clay-like soil on the roots.  If you've seen my opinions on bonsai soil and drainage, this won't surprise you.  Collected trees around here (north Alabama, USA) generally come up covered with a  cement-like red clay.  Not good in a pot!

    Root-pruning to maintain a bonsai consists of combing out the roots with a stick or fork-like rake, then trimming them back closer to the trunk, and repotting usually in the same pot again.  You can also clean out a small wedge-shaped section between the larger roots each repotting to gradually refresh the soil and roots closer inside the rootball.  As a general rule, I would say to remove enough roots so that the root ball is half of the pot volume, and fresh bonsai soil the other half.  This gives you plenty of room for roots to grow until next repotting time.

    The toughest part is knowing when to repot --- disturbing or pruning the roots at the wrong time will kill the tree; done at the right time, the tree will recover quickly and flourish.  For this, I'll describe 3 types of tree, and my experiences with each.
    First is the hardy deciduous tree.  This is one that loses its leaves over winter, then resprouts in spring (like elms, maples).  It is the easiest to tell when to repot.  In spring, the leaf buds will begin to swell.  That is the best time to root-prune and repot.  It doesn't need roots as critically without its leaves, and the spring growth surge that is about to begin will quickly regrow new roots to help it recover.
    I have taken many deciduous nursery plants and, at this bud-swelling time, chopped the root system back from a massive tangle to almost no roots at all, using hatchet and saw, removing any remaining soil with a jet of water, and repotting, with probably not a single loss of tree.
    Second is the evergreen, by which for this discussion I mean any winter-hardy outdoor tree which does _not_ lose its leaves in winter.  The discussion for deciduous trees applies here pretty much, except that it is not alway obvious when spring is arriving. You almost have to root- prune and repot early to make sure you catch it before it comes out of dormancy.
    Junipers are difficult for me to tell that they are about to grow.  Once you see the bright green new foliage, they're already growing.  At least with pines you will see the "candles" begin to lengthen, and that is their time.
    Thirdly is the "tropical", by which I mean a tree which is not winter hardy and should not be subjected to freezing, and usually must spend winter in a house or greenhouse.  These go thru growth and dormancy cycles much like hardy trees, but they are not as obvious.  To further confuse things, I have read that they should be repotted during a growth cycle, whereas I would have thought during dormancy or as they are coming out.  I have probably lost a few tropicals due to not recognizing the right time!

Precautions:
    - Be brave about root pruning for the first time.  Roots must be pruned for the tree's sake, and trees are great survivors.
    - Pick the right time for the  type of tree (easy, right?).
    - Keep out of direct sunlight (important!) for a few weeks after  repotting, until new growth is evident.  This indicates that new roots have regrown, also.
    - Do not fertilize until root system has recovered,  as indicated by the new growtht.  (There are different opinions, but not fertilizing is the safest side of the argument.)
    - Use clean, sharp tools to prune roots.  Do not use your branch-pruning tools to prune roots, as contact with soil and grit will tend to dull them, making them less useful for branch usage.
    - If a sudden deep freeze in the weather occurs before recovery after repotting, bring the trees indoors for the short period until it passes.