Taking cuttings is another method of increasing your stock, especially of desirable varieties or species which are rare.
Most cuttings are pretty easy with the proper environmental conditions.
Cuttings need:

1. A clean well-drained rooting medium,
2. A rooting hormone provided either naturally or by applying one,
3. A constant supply of moisture to make up for the lack of roots,
4. Sufficient light to allow the manufacture of food (softwood and semi hardwood),
5. Bottom heat in some cases, although almost all do better with it.

Rooting Medium

The medium must be sterile for good rooting percentages over a wide range of species. Some species appear to be immune to ever thing fungal in the soil and others are very prone to infection. Most roots also need air to form, as well as to minimize infection. You must balance the need for drainage against the need for a constant supply of moisture. Without automatic mist the medium must hold more water. A good place to start is 60% peat and 40% washed out river sand, we need to use the small particles, you can also add a little bit of perlite.

Rooting Hormones

There are a few types of rooting hormones on the market, all work well, make sure that you read the instructions carefully.  

Moisture

Without a root system cuttings rely on absorbing moisture through the stem and leaves. This works just fine if the medium is constantly wet and the humidity is high and it is cool, seventy to eighty degrees during daylight hours. This is the toughest aspect to achieve at home. You must reduce the rate of transpiration to protect the cuttings from wilt. You can do this by keeping them cool, keeping the leaf surfaces wet, and by reducing the leaf surface area. You can remove some leaves to reduce surface area so that only two or three small leaves remain, or you can as I prefer, remove the bottom leaves then cut the remaining leaves in half. After a while you get a feel for how much leaf area each species can support given your individual conditions.

Sufficient Light

It is thought that roots are stimulated in cuttings by high light levels, although I have not seen the studies to support it. But cuttings do root as a response to food moving down the stem in the phloem (remember your botany test, phloem and xylem) and stopping at the bottom cut. Finding no roots to store the food, it backs up at this point and changes take place in the presence of hormones to convert stem tissue to undifferentiated tissue (callus) to finally root tissue.

Bottom Heat

Automatic mist and bottom heat revolutionized the nursery industry around the time of the Second World War. Before that most production was by hardwood cuttings, which was slow and effective for a limited number of species. These days there are very few plants that cannot be propagated from cuttings and those that can't can usually be done by tissue culture.
Bottom heat acts as a stimulant for the production of roots as well as for faster root growth. In general bottom heat should be ten degrees hotter than the ambient air temperature, although any amount of bottom heat is useful.
Some species are more sensitive to heat than others. Most of the tropical’s I have grown in the past, liked it hot. Maples also root much faster when hot. It is thought that a few cultivars of Junipers actually prefer cooler temperatures once they callus.

Cuttings fall into three categories.

1. Softwood
2. Semi hardwood
3. Hardwood.

Softwood Cuttings

These cuttings are taken from new growth at the succulent stage. When a turgid plant tip snaps when bent at ninety degrees it is ripe for softwood cuttings. Most perennial cuttings are done this way, very few woody plants. In general low levels of IBA improve rooting but are not necessary.

Semi hardwood Cuttings

Most woody plant cuttings these days are from semi hardwoods. These cuttings are taken from wood that grew this year but is now firm and hard, with hardened leaves. It is well lignified at the base but still may have a soft tip that may or may not be removed. It is better to go by the feel of the wood instead of the calendar.
Many species will not root immediately, although most root over four to ten weeks. Some species will take a year or more to root after initially callusing in a few weeks. These I take off the heat at the end of the season and keep shady and cool until they do root
Many semi hardwoods respond favourably to wounding. The bottom of the cutting is sliced thinly through the cambium for about 10mm, taking out a sliver of bark without removing much wood. This is best done with the edge of sharp shears or a sharp knife to make a clean cut. This provides a surface for callus and hopefully roots development. It is also interesting to watch where roots develop on various species. If you wound, the roots will often form in a line along the callus tissue. Many cuttings form in a ring around the bottom of the cut (Chaenomeles), for some it is necessary to have a node at the bottom of the cut, other not (Clematis, Acer).

Hardwood Cuttings

Hardwood cuttings can be taken at any time of the year, but are usually taken late in the season or winter and stored in sand, sawdust, etc. They are cuttings from wood that has gone through an entire season of growth. The can be treated just like softwood cuttings in summer for some species, but most are done at the end of the season and either stored and planted in the spring after they callus or simply planted in open ground in the winter. With the advent of mist and bottom heat their popularity has diminished except for some commercial crops such as grapes.

And finally

Be Clean! To avoid fungal infections, a very serious problem with cuttings, keep everything as clean as possible. Think of it as handling meat in the kitchen. You must always remember where those hands have been. Use a ten percent bleach solution (nine parts water, one bleach) to clean everything including the flats, shears, your hands and the cuttings. After the cuttings are made, soak them in the bleach solution for fifteen to twenty minutes, rinse all of the bleach off thoroughly (it will oxidize the IBA), then dip them in hormone and stick.

Acknowledgment to Bonsai Primer.com

Growing Bonsais from Cuttings.