The most popular Chinese character typing method in the US and Mainland China is Pinyin input. Pinyin input simply involves typing on a standard QWERTY keyboard and spelling out the desired word in Pinyin, and then selecting the correct characters from a pop-up selection on the screen. Pinyin input requires only a knowledge of Pinyin spelling rules and use of a standard American keyboard. For an American second language learner, Pinyin input is the easiest method to learn.
There are many other methods of Chinese character input on computers, including Bopomofo, popular in Taiwan, and Canjie, popular in Taiwan and in heritage language communites world-wide. Read more about the different Chinese character input methods.
A demonstration of the Windows Pinyin IME. In Mandarin Chinese.