Whether you're connecting to vCenter or ESX(i) host themselves using the vSphere Client, majority of the requests are using vSphere API .... I say majority since I'm sure there are some undocumented ones that VMware is using behind the scene but in general, this is true. If you're performing an operation within vCenter and it applies to a host, vCenter will communicate that request to the host.
If you're new to APsI/SDK's: You may want to spend some time going into the 'Learning' section of the developer forums, there's lots of helpful information there: http://communities.vmware.com/community/developer/learn
Here is a great doc Getting Started with vSphere SDKs - Updated for SDK 4.1 release that has lots of information/explanation on how all this works, you should spend time going over it and then ask any follow up questions in the appropriate forums if you need further explanation.
VMware vExpert 2009
VMware ESX/ESXi scripts and resources at: http://engineering.ucsb.edu/~duonglt/vmware/
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Thank you very much for references.