There are many different input methods. In Windows, it's is called an IME (input method editor). You type several keys in the IME window, and it generates a Chinese character.
The simplest methods are based on phonetics, such as Pinyin system mentioned by Democoder, or Bopomofo (a similar system used in Taiwan, which has its own symbols). Since it's based on phonetics, it's very easy to learn. However, since there are many Chinese characters with the same pronuciation, it's very frequently that you need to choose the character you want from a list, sometimes a very long list. It slows down the typing speed. Furthermore, if you don't know how to pronounce a character, perhaps some rare characters, you won't be able to type it.
In current Windows, the phonetic base IMEs are improved with some techniques. The new IMEs can "guess" the character you want from the list, based on the previous characters you typed, so you don't need to choose every time. You only need to "correct" it when it guessed wrong. It's like the T9 text input method on cell phones, where you only need to press each key once and it tries to guess the word you want to type (such as "469" gives "how"). However, many people posting on a public forum don't bother to correct them, so now there are many forum posts with wrong characters, just like spelling error in an English post.
For professionals or any people who need to type fast, phonetics based methods are not sufficient. Sometimes professional typists need to type rare characters, and they don't have time to find out how to pronounce them (rare characters are common in people's names). These people normally use a "shape-based" method. In shape based methods, Chinese characters are splitted into many "base shapes," many of them are radicals, others are common shapes (such as a square shape, which means "mouth," appear in many different Chinese characters). Since there are hundreds of base shapes, the shapes need to be grouped into 26 groups (a little more or less) to make them compatible to English keyboards. The grouping are designed carefully to minimize the need to choose characters from a list. A good "shape based" method should let you input any Chinese character with less than 5 key strokes.
A good "shape based" method rarely need to choose characters, so you don't need to look at the screen. However, the base shapes and the grouping are complex, and it takes time to learn and memorize them. Many improvements to the shape based methods are made to ease the learning curve. For example, one famous input method uses 'O' key for the square shape, and another shape which pronunced as 'o'. This is designed to make it easier to memorize them. However, most people still don't bother to learn them, since phonetics based methods are "good enough."
It's a different story to input Chinese on a cell phone. Of course, Pinyin or Bopomofo based methods are still there. However, there are also simplified "shape based" input methods. These methods are not design for fast typing, but rather for the situation where you don't know the pronounciation of a rare character (you certainly need to input a lot of names on your cell phone
).