An implication may be either an entailment or a conditional statement; these are closely related but not quite the same thing.
A conditional statement is the result of a binary operation on propositions within a given context in a given logic. If and are propositions in some context, then so is the conditional statement , at least if the logic has a notion of conditional.
Notice that , , and are all statements in the object language (the language that we are talking about), whereas the hypothetical judgements and are statements in the metalanguage (the language that we are using to talk about the object language).
Depending on what logic one is using, might be anything, but it's probably not fair to consider it a conditional statement unless it is related to entailment as follows:
You can think of entailment as being an external hom (taking values in the poset of truth values) and the conditional as being an internal hom (taking values in the poset of propositions). In particular, we expect these to be related as in a closed category:
where is an appropriate constant statement (often satisfying , although not always, as in linear logic with for and for ).
Just as we may identify the internal and external hom in Set, so we may identify the entailment and conditional of truth values. In the Lab, we tend to write this as , a symbol that is variously used by other authors in place of , , and .
Although Heyting algebras were first developed as a way to discuss intuitionistic logic, they appear in other contexts; but their characterstic feature is that they have an operation analogous to the conditional operation in logic, usually called Heyting implication and denoted or . If you use and replace above with the Heyting algebra's partial order , then everything above applies.
In type theory