I'm sure there is a tutorial somewhere around here about this subject. But I am awful about finding those. Someone will come along with a link in a bit, I'm sure.
A cutting is just that, a cutting of a
growing part of the plant. Different plants have different times that cuttings are best taken, some in
spring from green new growth or some in fall from woody growth. It really depends on the plant you want a cutting from.
A
greenhouse is the best place to propagate cuttings. But a
greenhouse effect can be made with a simple clear plastic bag over the plant and held away from the plant itself with sticks or something similar.
You need a few inches of a plant that you remove all but the upper leaves from. My favorite medium is sterile potting soil. I poke a hole into the soil and stick the cutting into it firming the soil up close to the stem. You want to make sure that a few leaf joints (the places where you removed the lower leaves from) are under the soil. That is where new roots will begin to form. I stick 2-3 chopsticks into the soil around it and cover with a plastic bag. Secure the bag with a rubberband around the pot.
These are best kept moist most of the time and in a bright, location out of direct sun.
This works with most plants. But take several cuttings in case you get a cutting or two that don't take. The timing is different on different types of plants also. Some will be rooted in a few weeks, others might take a few months.
Some people also root plants in water, but I have never done this.