How is a conflict between aircraft with TCAS I and II resolved?
TCAS 1 will only give a Traffic Advisory (TA). The crew will lookout for the other aircraft and take evasive action if necessary. They also may contact ATC for instructions. They will follow ATC instructions unless they see that the instructions will bring them into the path of the other aircraft.  

TCAS II will give a traffic advisory first, then a Resolution Advisory (RA). The crew has to follow the resolution advisory and inform ATC of their maneuvering (e.g. TCAS climb, Big Jet 123). They are trained not to follow ATC instructions that oppose the TCAS RA. 

Note that TCAS II RA's are always in the vertical plane; TCAS II can not give instructions to turn.