What do the different ICAO Flight Rules types mean?

Currently in the US, only VFR and IFR types are used. Choose VFR if the entire flight is planned to be operated under VFR. Choose IFR if the entire flight is planned to be operated under IFR. 

If you plan to change the flight rules under which you the flight will be operated during the flight (ie: VFR from departure to intermediate point, then change to IFR) you should file two flight plans. 

FYI: The ICAO codes YFR and ZFR are not supported in the US but may be used in other countries. YFR means the flight initially will be operated under IFR, followed by one or more changes of flight rules. ZFR means the flight initially will be operated under VFR, followed by one or more changes of flight rules.