Is there any difference between coreference resolution and anaphora resolution? More generally, what is the difference between coreferences and anaphoras?
An obvious difference I see (or, rather, one Ruth Kempson saw in a 1977 textbook of hers), especially from a cross-theoretical perspective (i.e., leaving terminological issues aside as much as possible), is that 'anaphora' is an intra-linguistically determinable relation, whereas 'co-reference' necessarily requires access to 'extra-linguistic' information. By definition, co-reference always implies 'identity of reference', whereas anaphora does not (I ignore here how the term anaphor - not anaphora - is used in Principle A of Chomsky's Binding Theory). In other words, the relation called anaphora may (but need not) hold of expressions that are supposed to have the same 'sense' (either in themselves, or only via context-dependent synonymy) but may still have different referents (and usually do), whereas co-reference does not depend on 'identity of sense' at all, but, of course, does strictly require co-referential terms to refer to the same extra-linguistic entity. This is most obvious, perhaps, in the use of one in John has a BMW and his girlfriend has one, too, where two different BMW's are being referred to, although similar examples can be built with all the other pro-forms (i.e., so, do, do so, that/those, mine, yours, etc., cf. John finished his thesis in 1990 and Bill did so a year later, Your thesis is better than mine, etc.). On the contrary, co-referential terms, of course, may (but, again, need not) have completely different 'senses', and yet, by definition, they must refer to the same extralinguistic entity. Thus, in My wife is sad because she misses her father, My wife and she have different 'senses', but may still have the same 'referent' (and that's how the sentence is interpreted by default, in the absence of further context); on the contrary, in She is sad because she misses her father, both tokens of she cannot but have the same 'sense', but they still may (or may not) be co-referential, although, in the absence of further context, they will be interpreted as co-referential; in John lost his temper, finally, John and his must be co-referential. In view of that, you may expect huge differences in the resources and computational strategies respectively needed to identify 'anaphoric' and 'co-referential' expressions; typically, determining whether co-reference, an extra-linguistically determined property, holds or not between any two expressions will require much more 'pragmatic' information ('knowledge of the world') than identifying pairs of expressions in the strictly intra-linguistic relation of 'anaphora'.
The answer by Tim Osborne is very comprehensive and accurate. But perhaps things might be easier when looked at through usage in text linguistics.
Although, co-reference and anaphora can be distinct, they are often used interchangeably by linguists. This is partly because by far the most studied type of anaphoric reference is co-referential. So you could say that co-reference is one of the processes through which anaphora is encoded. It is not just a type of anaphora because co-reference can also be cataphoric (although some people use anaphora to mean reference in both directions).
So the difference between anaphora and correference resolution follows from the above.
Coreference anaphora is just one type of anaphora. Following George Lakoff, syntacticians distinguish between identity-of-reference anaphora (which includes definite pronominalization) and identity-of-sense anaphora (which includes indefinite anaphora with "one"). So as a first approximation, the answer to the question is that anaphora includes indefinite anaphora, or that is to say, identity-of-sense anaphora, as well as coreferential anaphora.
For instance, in "He ate a green apple, but I ate a red one", in interpretation, "one" means "apple", but it is the meanings that are the same between antecedent ("apple") and pro-form ("one"), not the individuals. He and I did not eat the same apple.
The question is difficult to answer due to its brevity and the lack of context in which the terms coreference and anaphora appear. However, I can provide some orientation that should help increase understanding of how these terms are used.
The term coreference denotes a situation where two or more expressions refer to the same one entity in the world of discourse. In a situation where a speaker says
(1) Look at that man over there; he is wearing a funny hat.
the phrase that man refers to the actual man in the discourse situation of the speaker, and the pronoun he refers to the phrase that man (its antecedent), so it too refers to the actual man in the discourse situation, although indirectly. Thus the two expressions, that man and he, are coreferential; they refer to the same one entity in the discourse situation.
Words or phrases that point to other words or phrases are anaphors. The term anaphora denotes the actual act of pointing (anaphora is technically already plural, so anaphoras is incorrect). Any time a given expression (word, phrase, or some combination of words) points to some linguistic expression (its antecedent), it is an anaphor (or cataphor). Pronouns are typical anaphors, but many other types of expressions can also be construed as anaphors, e.g. verbal expressions (do so, do it), adverbs (there), preposition-like combinations (thereof, therefore), etc. Any time one of these expressions occurs in speech or writing, anaphora is present. The anaphor is pointing to some expression in the linguistic or situational context.
The main point of confusion that arises in this area concerns the presence of anaphora without coreference. Consider the following sentence in this regard:
(2) No man said he was hungry.
Anaphora is present in this sentence, since the pronoun he, an anaphor, is pointing back at its antecedent no man. But the antecedent no man does not pick out an actual entity in the discourse world. The expression no man cannot be construed as referring to anything at all. What this means is that coreference is not present, but anaphora is.
When one spends some time probing these distinctions, one quickly sees that the terminology is not clear, and often the terms are used inconsistely or unclearly. But that is understandable, since the notions at play are rather fuzzy, even for many seasoned linguists.
An appeal to caution: the term anaphor has a special meaning in Government and Binding Theory (GB) and the Minimalist Program (MP). These frameworks of syntax use the term anaphor to denote what most of linguistics in general calls a reflexive pronoun (or a reciprocal pronoun). Thus in GB/MP, words like himself, herself, themselves, itself, each other, etc. are called anaphors. The study of syntax and grammar in general uses the term anaphor (and cataphor) more broadly, to denote any expression that points to some other linguistic unit or entity in discourse.
Finally, I'm not sure what is meant by "coreference resolution" and "anaphora resolution". I assume those terms are meant to denote situations in which an anaphor successfully finds its antecedent (or postcedent) in context. Anaphora resolution would, then, fail to occur if a given pronoun appears for which it is not clear what its antecedent or postcedent is supposed to be, e.g. Fred and Tom said he was hungry. Does he refer to Fred, Tom, or to someone else entirely.