Removing undesirable parts the plant and reducing other parts will improve light and reduce competition benefitting future growth.

Pruning enhances the inherent characteristics of the rose, often with bigger and better growth and flowers.

Pruning improves future growth.
Pruning doesn't change growing habits.

When pruning consider the natural habit of the plant. Prune to enhance a habit – not change it.

What to prune depends what the natural growing habit of the plant is (and sometimes what you want it to do). If the bush produces long stemmed flowers and you want lots of long stemmed roses, you prune it to produce long stemmed flowers. If the rose has a bushy habit with lots of flowers on shorter stems you should prune it to produce that kind of growth. Climbing roses and weeping standard roses are treated differently to bushes because you want to encourage flowers along the branches as well as encouraging new branches.

(Note: growth characteristics are defined by the bud characteristcs. It's futile to try to force short growth when the buds produce long growth shoots. (I.e. you can't keep a tall rose down - e.g. you can't get "Mr lincoln" to grow like an "Iceberg".) If short growth is required choose a rose where the buds make short shoots.)

bud/shoot competition
The buds on a branch share the resources of the branch. The harder a branch is cut back, the more resources are available to the remaining buds. The buds use these to grow and then flower.

Importance of light.
Shoots grow toward a light space and their leaves use that light. Shoots that receive more sunlight are more powerful than those that receive less light.

To cut or not to cut.
Roses have different styles and growth structures and have different considerations when pruning. These are discussed below in How - different rose styles and When - repeat flowering / once flowering. But if you get it wrong it's not fatal. You might stuff up some flowering or temporarily promote the wrong shape, but buds will regrow, the stonger growth will take over and you can give it another go. (Things that go wrong after pruning are usually a consequence of growing issues not directly associated with the cuts. So to all those husbands that get blamed after a prune - it's not your fault.)

Where to cut - it's about the buds.
Growth is defined by the buds developing into new shoots and is governed by light and plant resources available to the bud.
Buds are distributed periodically along a branch and are the main points of consideration when pruning.
No growth occurs where there are no buds. Cut close to just above a bud to minimise the amount of inactive branch beyond the bud.
Choose an outward pointing bud.
Buds like light. Even though there might be a light space for them to grow into now, later it may be shaded by the leaves of other growing shoots. Buds are happier and grow stronger shoots if they can play in the sun, so prune to a bud facing outwards into sunlight.

There are several buds on a branch using and competing for the resources of the branch. With fewer buds on the branch more resources are available to the remaining buds resulting in a longer shoot growth. (See How - different rose styles below.)
Summarising: the harder you cut the longer the shoot.

Which branches to prune.
Choose to remove branches that are:
- Thin/spindly. Their buds don't get much to grow on.
- Growing inwards. Their buds won't get sufficient light.
- Crossing and rubbing against each other. Danger of damage to the branch.
Choose to keep branches that provide a happy bud community with plenty of sunlight and branch resources.

Pruning warning #1;
- don't prune a once a year flowering rose (e.g. banksia roses) until after flowering.
Otherwise it might not flower until next year, or ever if you keep doing it.

Pruning warning #2;
- wait until after the danger of frosts.
Frosts may damage new growth.
When this is depends on where the rose lives.

Other than that, you can prune after any flowering period.
Buds grow into shoots flowers at the end and dormand buds in the leaf axils. When the shoot has flowered those dormant buds become active and again grow into shoots with flowers at their end (repeat flowering types).
So when a branch has flowered you can cut to bring out the best in the remaining buds. Most likely there will be some flowers developing and some which have finished. Whether you cut some or all is your choice - repeat flowering roses will reflower anyway.
Often, the more you cut the more flowers you get!

Buds define the growth characteristics of the rose plant, using resources from their point of origin on the plant and sunlight to develope shoots that give the rose its growth and flowers.
Buds on a branch grow shoots that compete for the resources available to the branch and are empowered by sunlight.

Pruning manages potential growth by cutting branches to increase the competitive advantage of the remaining buds.

Warning: don't prune below the graft.
(As if you would.)

How hard to prune depends on the natural habit of the rose.

Hard pruning / light pruning
Generally hard pruning will produce longer strong growth and is very useful for cut-flower types, whereas light pruning produces a more bushy gartenesque affect.

Hard prune

- Remove weaker growth
- cut stronger branches back hard to just a few buds.

This concentrates resources into just a few buds, at the same time allowing maximum light to power them.

Often used to:
- encourage long stemmed flowers,
- rejuvenate/correct growth.
Particulary useful when trying to correct one-sided growth.
How far to cut back is up to you. 50% or more, ankle height, is all O.K.

(Note: Cutting back hard creates longer stronger shoots. It doesn't make the growth short.)
Light prune
Roses that produce a bushy type of growth benefit from a light pruning.
Light pruning allows the buds on the existing structure to provide growth rather than concentrating all the resources into just a few. It gives a "softer" bushy look than with hard pruning.

- Remove weaker growth,
- open the centre to light by removing inward growth,
- optionally reduce other branch growth,
- encourage outward growth. Usually retained branches are not cut back by more than 30%.

Pruning different rose styles

Long stemmed roses

e.g. hybrid tea roses, cut-flower roses, etc

Growth buds produce long shoots, often with a large flower or sometimes a small group of large flowers at the end of the end.

These roses benefit from a hard pruning to produce new strong growth.
They can be pruned multiple times during the year as long as it's warm enough for new shoots to grow i.e. from the beginning of Spring to Autumn.

Bushy/shrubby roses

e.g. Most floribunda, most David Austin/English roses, most old world roses, etc

Usually "bushier" than long stemmed roses, often producing multiple flowers on the end of a shoot.
The flowering stems are often shorter but the plants can be small or very big, depending on which rose it is.

These are mostly pruned with a light pruning, to get best from the existing growth as well as encouraging new shoots. Remove the bad bits and keep the good bits.
The aim is to enhance vigour while retaining shape.

Climbing roses

Flowers all over.
Climbing roses have shoots that make long branches with flowers at the end, but also with lots of buds distributed in the leaf axils. These buds will develope into shoots, also with the potential of flowers at their ends. If they all develop at the same time they are effectively sharing in the resources of the host branch. They will only have the ability to grow a short distance before they flower, producing flowers along the branches.
Allow light in.
Pruning climbing roses allows this to happen by leaving some shoots long while removing undesirable growth. Thin growth to allow light in to the base and also to benefit desirable buds.
Leave some shoots/branches long.
Apart from leaving some new growth long, there may be older long branches that have previously had flowers on short growth along the branch that may be kept. (Usually for the sake of retaining a balanced shape.) The short shoots may be cut back somewhat, leaving a few buds on them that will again produce flowers on short growth.

Training branches horizontally effectively allows light to reach most of the buds on the branch equally. They all effectively grow at the same time (sharing the resources of the branch) and flower together with a massed display.

Warning. There are some very spectacular once a year flowering climbing roses. Remember not to prune them until after they have flowered.

Standard roses

Standardised roses may be of the long stemmed or bushy type. If bushy they can be light pruned. If long stemmed it is best not to prune too hard.
Pruning tall standards
Long lanky growth is often not desirable on a standardised rose, but often forgiven because of the beauty of the flowers.
Pruning a branch hard concentrates resources into a few buds and results in strong but long shoots. Cutting a branch not so hard helps to produce less vigorous shoots. The tall roses will be still be tall but bushier.
(Note. You can't make tall small by cutting back hard. It will only produce longer shoots.)

Weeping standard roses

Prune similar to climbing roses. Keep some branches long to encourage flowering along the branches.
(Note the warning to prune once a year flowerers after flowering.)