It seems to be a common misconception that an American can just pick up and move to the UK just because they’d like to, there's a "special friendship" and we share the English language.
This is entirely untrue, you need to qualify for a visa to move to the UK!
The main ways that US nationals come to the UK are as visitors, as fiancee’s/spouses/civil partners of UK or EEA nationals, or tied to studies, employment or investment. With all of these categories, it is essential to prove that you will not need to be supported by state funds (ie welfare) once you get here, and that you have somewhere to live. It is important to note that not just *any* job can get you a work permit, and a casual relationship (ie one where you have not been living together and do not yet intend to marry) will not give you enough ties to emigrate. The Visa section will give some detail on the different categories, and give links to government departments which contain the information you will need.
If you have questions about the process, please post them on our dedicated UK Visa discussion board.