Meditation, especially when done fairly regularly, is Good For You. It will reduce high blood pressure and other stress-related effects, including in many cases unnaturally high cholesterol content in the blood. It will make it easier to fall asleep and wake up, and most people will feel better with themselves and life. Recommended dosage is half an hour a day, though some want more and most will notice effect from less.
However, in a few people meditation can be harmful, especially the more advanced forms. This is people who are psychotic or on the verge of psychosis. They may experience that the walls of the self are partially removed by meditation, and scary stuff jumps in. I advice that anyone with severe personality problems in the family do not go deeply into meditation without professional handholding. (Medical profession, not religious profession, please.)
The most common (and easy to reach) state of meditation just happens to coincide with a state of mind which can be measured in EEG (Electro EncephaloGram) as "alpha waves". These brainwaves are fairly stable and will typically span across both halves of the brain. They come with a relaxed state of mind and body, where the person is still conscious. Normal humans tend to pass through this stage just before falling asleep. Sometimes it can be provoked by assuming a restful position (such as sitting idly but upright in a chair). Then close your eyes and roll them lightly up, as if trying to look out through a point above the base of your nose. Do not frown; if anything, lift your brows, then relax the muscles. The accompanying state of mind includes awareness but not "hard" thinking, and in its pure form there will be little or no pictures or internal speech-thought. Maths is highly incompatible with the alpha state, as is worry and fear. Going consciously into the alpha state is to most normal people a positive experience, and it becomes easier the more you do it.
There are many ways to prepare or train oneself in meditation. The point is really to get used to one's own awareness. Awareness not of the senses but of the mind itself. For introverted people this often comes more naturally, while it can be harder for a very extrovert and restless person. In eastern religious meditation it is common to use a "mantra", a word or phrase that is repeated slowly (often silently) as an anchor for the awareness. Another approach is to look at a symmetric pattern, for instance a "mandala", or indeed any object that easily confines the awareness (circles and squares are well suited). -In western religions, meditation is usually associated with prayer. After praying, if you "listen" (without straining) for any sense of contact from Above, you may enter the meditative state of mind. To some people this can be a deeply religious experience. ("Practicing the presence of God".)
A simple and religiously neutral technique is counting slowly upward to ten, or even just four. For best effect, most people will want to time this with their breathing, for instance "one (breathe in), two (breathe out)". The point is not to count anything, and definitely not to reach ten as fast as possible. Slow is good. Usually when this works, you will find that you are counting slower and slower as your breathing slows down and your body relaxes. It will then stabilize at this level.
Normally this does not happen all at once. The human mind is like a flock of sheep (thoughts) running all over the place. You will most likely find stray thoughts running off with your concentration most of the time. This is normal and to be expected. So don't get angry or bitter or disappointed. It is OK. Just, when you notice it, start counting again. If you don't remember how far you were, start at 1 again. Remember, if you chase your thoughts they will run all over the place making a lot of noise. Go peacefully in front of them and they will eventually start following you ... though it may take months or years. The goal is not as important here as the road. It is the practice of becoming aware of mind / brain states.
There are other brain states that are possible to reach, though this usually takes a thorough practice in the alpha state. A quicker way is available through biofeedback, though one might argue that the discipline of "manually" practicing meditation can be a bonus. Biofeedback (and, some claim, "brain machines" / tapes) can be a help to get started if the traditional techniques don't work.
Some people who are otherwise mentally healthy may experience "paranormal" effects more often after meditating regularly. These can vary from successfully knowing what others think, to unlikely quirks of fate, to the more suspect effects such as seeing great white lights or feeling intense physical pleasure (ecstacy). A feeling of weightlessness is quite common and natural, and is not a sign that you are actually levitating. The reason for these "magic" effects are not fully known, and some people will go clear of them totally. Generally, if you're otherwise mentally stable, you can just ignore them and concentrate on your meditation. If you can't ignore them, and find that you are probably someone big in the cosmic hierarchy, you are advised to stop meditating immediately. But normally this should not happen, and you should be in full control (if anything, more than before) all of the time.