If Δ ⊢ Φ, then Δ ⊨ Φ
has an implicit universal quantification to it:
For all Δ, Φ: if Δ ⊢ Φ, then Δ ⊨ Φ
Unsoundness of a proof system then means
Not for all Δ, Φ: if Δ ⊢ Φ, then Δ ⊨ Φ
This is equivalent to
There exist Δ, Φ such that not: if Δ ⊢ Φ, then Δ ⊨ Φ
which is in turn equivalent to
There exist Δ, Φ such that Δ ⊢ Φ but not Δ ⊨ Φ
That is, an unsound proof system produces proofs for inferences that are not actually valid. Of course, this makes the proof system rather useless, since you want a proof system as a device to show that an inference holds, but in an unsound proof system the situation is precisely that you do not have the guarantee that the proved inference actually holds.
Since unsound proof systems are not very useful, you won't commonly run into them when studying logic. I don't know off the top of my mind of any real-life example, but of course, it can well happen (and most likely has happend throughout history) that the developer of the proof system had the intention of a sound proof system but made a mistake in the design of the rules and had them not adequately reflect the semantics, so that it later turned out to actually be unsound (and didn't make it to popularity for that reason).
And of course, it is easily possible to just ad hoc define some random nonsensical proof system on purpose, e.g., by inventing a rule that says
or the like. Any proof system with such a rule included will be unsound because obviously A ⊭ ¬A. Note that this again encompasses an implicit universal quantification: Adding this rule results in A ⊢ ¬A for all formulas A, while we do not have for all formulas that A ⊨ ¬A, which is why the rule is unsound; though there may be some formula instances for which the inference does hold (e.g. with A = ⊥). And of course, the proof system may well encompass other rules that are sound and produce proofs for inferences that hold for any instance of formulas. So unsoundness doesn't automatically mean that all of its proofs are wrong.
Neither does it mean that it proves any contradictory formulas. A formula being valid means that it is true in all structures; if a proof system proves, say, ⊢ A v B --> A, it is unsound because A v B --> A is not valid, i.e. not true in all structures. But neither is it contradictory; there do exist structures and instances of A and B in which A v B --> A does hold (e.g. any structure in which A is true).
And unsoundness doesn't automatically make the proof system inconsistent: A proof system is insoncistent iff it proves both A and ¬A for some formula A, that is, if it proves a contradiction. Suppose A is valid (hence ¬A is contradictory), and the proof system proves ¬A but not A. Then the proof system is unsound, because with ¬A it proves a formula that not actually valid, but it is not inconsistent, because it doesn't prove A which would be necessary to derive a contradiction.
For the combination complete + unsound, see also here: What is true about a proof system that is complete but not sound?
Unsoundness means that the system proves inferences that aren't actually valid, i.e. there exist sequents Δ ⊢ Φ for which Δ ⊭ Φ (= there exist structures in which all formulas in Δ are true but Φ is false).
Unsoundness makes a proof system somewhat pointless and will usually only happen by accident or for demonstrative purposes.