What is the relationship between C-command, binding and Co-reference? Does it mean that C-command is the precondition of binding? Can somebody help me get clearer about these three terms?
In its most basic incarnation, binding is a combination of c-commanding and coreference.
In other words, in a sentence like Sam talked to herself, Sam c-commands herself and is co-referential to it (the two phrases refer to the same person). Therefore Sam binds herself.
But in Sam talked to him they don't co-reference, and in Alex talked to Sam, and James talked to her too they don't c-command. So in these sentences, Sam does not bind her.
Notably, in the first sentence we see a reflexive herself, and in the other two we see a non-reflexive her. Which brings us to the whole point of "binding": we can say reflexives must be bound, and non-reflexive personal pronouns can't be.
There are a few concepts being conflated here. C-Command is a tree-geometric relationship that describes two nodes relation to each other. @Draconis is correct in the sense that C-Command (specifically asymmetric c-command) is necessary but not sufficient condition for binding. The examples s/he offers however are incorrect (In all cases the subject is c-commanding the predicate), the issue becomes one of how that tree geometric structure is processed semantically (see Heim and Kratzer 1998). Binding is related to scope (which is defined by c-command) where a quantifier (in the Lambda-calc sense) binds a variable--in the cases above the DP subject is quantifier that is binding a pronoun (a variable). Whether that can be a reflexive binding or not is governed by Chomsky's Principle A B or C respectively.
The last comment Draconis makes: " Notably, in the first sentence we see a reflexive herself, and in the other two we see a non-reflexive her. Which brings us to the whole point of "binding": we can say reflexives must be bound, and non-reflexive personal pronouns can't be." Is not really true either. Binding is more than pronoun reference, it is also crucial in sentences with traces (eg What did you eat t? and other A-bar movement scenarios). S/he seems to be confusing coreferentiality (or co-indexing) with reflexivity constraints (eg Mary gave sally her favorite toy) where her can be referential to either Mary or Sally. Ellipsis phenomena seem to illustrate this as well:
Bill washed his car and John did too
There are two readings here (strict/sloppy in in the literature) (i) John washed John's car and Bill washed Bill's car (ii) John washed John's car and Bill washed Johns car too
Another example you may want to consider are islands/crossover effects:
Take: (iii) John wonders whether Bill ate his food.
Wh-movement here across a CP boundary still creates a c-command relationship but binding is blocked:
*(iv) Who does John wonder whether t ate his food?