American and other foreign names are transliterated into Chinese characters when written in Chinese. The transliteration is based usually based on the sound of the name in the original language and not on the meaning of the name. Common names in English have a standard transliteration, so for instance...
This means that an American name, or any other name coming from any language other than perhaps Japanese or Korean, will have no particular, special, or otherwise interesting meaning in Chinese, except perhaps by pure coincidence.
The characters chosen for the transliteration sometimes have a pleasant meaning, but often have no particular meaning at all. Many characters are reserved for transliterating foreign names, and are used for little else, so trying to find out what the transliteration of your name in Chinese means is usually a futile exercise.
However, foreigner often adopt a Chinese name, or are given one by a Chinese teacher or friend. These names can be anything you want, though since naming conventions in Chinese are very culturally sensitive, it's best to get some help from a Chinese speaker rather than simply try to create one on your own. You may accidentally create something that has an undesirable or unappealing meaning or connotation and only find out when it's too late.
For people studying Chinese, receiving a name is almost a rite of passage. Some teachers insist on giving all students a Chinese name from the first day of class, while others don't bother with it at all unless the student asks. Common names in English that have standard transliterations are usually preferred, but if you have an unusual name, your teacher or friend will attempt to create something with a similar sound similar in Chinese, or perhaps give up on that idea and create something with a pleasant meaning.
If you have a Chinese name, don't try to use it for official purposes in China. When applying for a visa, opening a bank account, or anything else involving the government or law, use what is on your passport. The Chinese government can be very picky about this, so keep to what your passport says exactly.
There have been over the years complaints about the transliteration of foreign names into English. One famous example was Kennedy, which was originally transliterated as 啃泥的 (kěnníde), meaning "gnaw mud person". It was then changed, for obvious reasons, to 肯尼迪 (kěnnídí), meaning "willing Buddhist nun enlighten".
A handful of foreign names, mostly place names, were given characters based on the meaning of the name in the original language instead of being transliterated. For instance, Oxford in Chinese is 牛津 (niújīn), the meaning preserved but of course the sound completely different.By contrast, Hyde Park is 海德公园 (hăidé gōngyuán), meaning "sea virtue park", a very nice combination of maintaining the sound while creating an appealing meaning.
You have a lot of choices for your name in Chinese, since there is considerable flexibility. If a standard transliteration exists, it is best to use that, at least until your Chinese language skills are strong enough to explain why you chose something else. If no standard transliteration exists, get suggestions from Chinese-speaking friends, coworkers, or teachers. And don't forget, you can always change it later. The Taiwanese are well known for changing their names based on the recommendations of fortune tellers and the circumstances in their lives. You too can do so should you want to.