One of the core pieces of understanding Buddhism is practicing meditation. Meditation is one of the most important things you can do to improve yourself. It is also generally poorly understood. The purpose of this is to explain meditation to someone who probably knows nothing about it, and to improve my terrible writing skills (lol). To that point, if there is anything unclear or poorly explained, please let me know.
Anyone can learn to meditate. It's not difficult, although it does take a lot of effort. One of the nice things about meditation is that even if you don't spend much time on it, the time you do spend will still help you improve yourself. There are many benefits to meditation, some of which are improved concentration, relaxation, and mental control. You will also learn things about yourself that you would never know any other way. You can even try meditating as you read this.
Before I explain what to do, I will explain a little of what meditation is and isn't. The purpose of meditation is to turn your full attention on your mind and remove the mental formations and barriers that we acquire over our lifetimes. It is not learning to levitate. It is not going into a trance. You will be fully awake and aware the entire time.
The first step to meditation is learning how to concentrate. You must learn to concentrate so that you can concentrate on your thoughts so they don't slip by. The easiest way to learn to concentrate is by fixing your attention on an object. Some people use a clock, beads, a color, a spot on the wall, but the easiest object to use is your breathing. You will always be breathing (or you won't be alive!) and you will have it with you everywhere you go.
First find a comfortable position to sit in. You might have some notion of a statue sitting cross legged making an odd gesture with its fingers. You don't need to sit strangely; you can meditate sitting in a chair. Your seat needs to be comfortable enough where you won't fidget, but firm enough that you won't fall asleep. Sit up straight, relax but keep your body firm, and put your hands in your lap so they are supported by the rest of your body. You should be someplace quiet and free of distractions.
Take three deep breaths, in through your nose, out through your nose. Just relax it. Breathe in and notice how the air feels inside your nose. Feel the air coming into your body. Breathe out and take note of the air against your nose. Breathe normally and calmly. Try not to think of anything but your breathing. When you think of something, push it aside and focus on your breath. In and out.
To keep focus on your breath, you can try counting breaths. Breathe in, breathe out, 1. Breathe in, breathe out, 2. Count to 10, then backwards to 1. If you miss a number, start over. You can breathing in and saying "in," then breathing out and saying "out" You can think of any number of different ways to do this. The purpose of it is to maintain your focus. The idea here is to focus your attention on your breathing and nothing else.
Soon you will begin to notice the nuances of your breathing you've ignored your entire life. You'll inhale and notice the tiny moment of joy before you exhale. You'll notice the tiny click in your throat after you exhale and begin to inhale again. You will notice the way your chest rises and falls as you breathe. Notice how your breath is constant. If you breathe in and hold it, you'll feel the tension start to rise in your body and the need to exhale. Calmly breathe in and out.
Focus your attention on the sensation of air flowing through your nostrils. This will be your focus point. Whenever you need to concentrate, bring your mind there. You can do this any place at any time. You can turn this focus on anything, but the purpose of meditation is to focus on your mind. Eventually you'll be aware of your breathing without controlling it.
Without being too specific, the general idea here is that when you focus your attention on your mind, there will be thoughts and ideas that come to your attention. Some of these are harmful and by dealing with them, having a conversation with them, reasoning with them, you will remove a piece of suffering from your mind and become happier. By doing this you will gain an understanding of your mind and the world. Remember to use thoughts and not words to reason things in your mind.
I've never actually explained this to anyone before, so hopefully I didn't leave any important steps or ideas out. If it seems odd, the best advice I can give you is to try it yourself. Meditation is all about finding things out for yourself.