Most territories have only one Performing Rights Organization (for example, Canada has SOCAN, Australia has APRA, etc.) that is in charge of the royalty collection for royalties generated in that territory.
The United States, however, operates a little differently. In the United States, there are actually four different PRO's. The American PRO's are ASCAP, BMI, SESAC, and GMR. Anyone can join ASCAP and BMI, whereas SESAC and GMR are invitation only.
Since it is the responsibility of the PRO's to collect the royalties generated in their territory, this means that there are multiple PRO's collecting the royalties in the US.
When you are an international writer and are asked to choose "US Representation", it is just asking you which PRO you would like to have collect your royalties in the US. You will only be able to choose between ASCAP and BMI, as the others are invite only.