All of the other posters are pretty much correct and, for an infantryman (I'm assuming from your user name that you are), there are at least 4 jobs I can remember ...
A slot with one of the Parachute Regt Bns (Coy Comd or 2IC, I think)
A slot with 16 Air Asslt Bde staff
A post at Warminster (HQ Director of Infantry if memory serves)
A post on the staff at RMA Sandhurst
The first 2 will require you to be airborne-qualified (Ranger/Air Asslt would be nice to have but not essential) but all are top-rated slots that you'll enjoy; as with us going the other way, the working environment will seem different but there'll be lots of common ground, particularly with ops the way they are. Expect to deploy in the first 2 jobs if your assignment coincides with a deployment (though your Embassy staff have the final say, as I found out). The Advanced Command and Staff Course (ACSC) als takes US students and has US Directing Staff.
Two final points:
1. Don't cripple your career. If any of these would get in the way of becoming Branch-qualified or doing CGSC, then delay them.
2. The post of Liaison Officer is different to these (these are Exchange Officer jobs). LNOs are usually Lt Cols and still work directly for their parent MoD/DoD in a given field, like Force Development. With these jobs, you are embedded fully and only a few bits of home nation policy (like UCMJ) will still apply to you.
Good luck in whatever you choose to do - if you come, you'll have a ball.
Consider making stateside purchases before you leave. In general, clothing is expensive and sizes and fit are somewhat different. Bring warm clothing and lots of sweaters. Housing (to include most quarters) are poorly insulated. Good raincoats, boots and umbrellas are a necessity. The British tend to dress in grays, dark blues, browns, and blacks - you'll rarely see them in loud sports clothes. They feel a sports jacket and tie is something to wear to a sporting event. Any social occasion or TDY means a business suit although slacks and a blazer may be worn to more informal occasions. Ask your sponsor for specific details.
Men should bring at a minimum (in addition to your tuxedo) two conservative business suits (one gray, one blue), one dark-colored sports jacket, and a number of shirts and ties to match.
Ladies fashions differ somewhat from those in the States. British women tend to "dress up" a bit more than the casual American. Wives should ship one or two evening gowns for the many formal functions they will attend in either the Officer's or Sergeant's Mess. Have your spouse correspond with your sponsor's spouse. Shoes that fit seem to be a problem for most women; narrow sizes are very hard to find.
Play clothing selection is poor. Although you can shop in the Air Force Base Exchanges, your choices will be limited. Consider purchasing clothing such as jeans, sweaters, and shirts in growing sizes prior to departure.