What are the rules for doing so?
To be very precise, the rules governing an American visiting the UK are published at Appendix V. These rules were put into place by an act of Parliament last year.
Briefly, they say that a US citizen does not require an entry clearance, just a passport. In order to determine if you are a genuine visitor, you will have a landing interview with an Immigration Officer. As to what the IO will ask you, we already have an answer for thatanswer for that. I suggest you read over it. If things go very, very badly, you will be bounced. You can read this articlethis article prepared for an American who had that happen.
When your landing interview is successful, you will get a 'leave to enter' stamp in your passport. Someone has already asked what this means in a legal sense and we have an answer herewe have an answer here.
However, do not wish to have difficulty with airport authorities for leaving and returning
Reading the articles mentioned above will give you detailed information about what to expect, and you will (or should) have a smooth transit through UK immigration controls.