Only for my understanding. The ESXi and the VSphere Client (V5.1) are free?
Yes ESXi is the Free Edition and it has certain number of limitation like maxmim 32 GB RAM, read-only to API's etc..
If you want to use the advance feautre of vSphere like vMotion, HA, DRS etc.. you need to go for the PAID edition of ESXi and you need to have vCenter server license separately
vSphere Client is the Free Managemnet tool to access the ESXi host and vCenter server