In analysing arguments, what are the differences between claims, support and premises?
Formally (see, for example, Campbell and Huxman)


A claim is an assertion.
A reason is the justification for making the claim.
Support is the evidence underlying your claim.
An argument consists of a claim + reason + support.


Let me give you an example of my argument about democracy in the Philippines.


Claim: The Philippines' young, vibrant democracy is facing real challenges in the 21st century ...
Reason: ... because the rule of law is threatened by endemic corruption,  extrajudicial killings and an armed secessionist movement in the south.
Support: Statements about the democracy and the need for the respect of the rule of law, the nature of corruption and its effect on democratic institutions, the moral problem of extrajudicial killings, etc.


Formally (see Aristotle's Organon), a premise is a statement from which a conclusion in drawn. This is nicely demonstrated in classic syllogisms:

Premise A: All men are liars.
Premise B: Socrates is a man.
===========================================
Conclusion C: Therefore, Socrates is a liar


In the syllogism above, A and B are premises. From both premises, the conclusion C follows.

Premises, by themselves, are virtually meaningless because they act as unreasoned or unsupported claims. For example, if I made the premise, 

"The Philippines' young, vibrant democracy is facing real challenges in the 21st century."


and left it there, you would ask such questions as 


what are your reasons for this statement?
what evidence do you have to support this premise?
what are the implications of this claim?