Unjust enrichment is an equitable theory on which a court can require someone who has received a benefit, even in the absence of a contract, to pay for the value of what he received.
Imagine a fellow who lives next door to someone who has contracted with some painters to paint her house. The painters arrive but instead of painting HER house, they paint HIS house. He watches and does nothing to stop them. At the end of the day, they say, well, we've painted your house, pay us. And the fellow says, I don't have to pay you, you painted the wrong house.
The painter then sues the fellow for payment on a theory of unjust enrichment. The idea is that the fellow received a benefit and it would unjust to allow him to be enriched by the receipt of that benefit, even though he didn't have a contractual obligation to pay the painters.
I'm not giving you legal advice, as I'm not licensed to practice law in Ohio. I'm just explaining the general principle of unjust enrichment. If you need legal advice, please consult a lawyer who holds Ohio licensure.
As my colleague above noted, it's a theory that people can use to file a lawsuit in the absence of an actual contract. Sometimes they tack this claim on to a lawsuit even if there is a contract. It's generally used where someone has provided a service that benefitted another (usually by agreement) where they have not yet been compensated for the service. If you have been sued, you will need to respond to the lawsuit. If not, the court will grant the plaintiff a judgment by default and then they will try and collect with bank sweeps and garnishments. I would recommend contacting a lawyer if you need further assistance.
As the other attorneys have explained unjust enrichment, I won't restate the concept.
I will restate that the only way to insure the sub does not sue you or attempt to lien the home is to demand the contractor secures a release of liens from all of the subs and suppliers.
Now, in Ohio, there are special rules with regard to residential work that will make it unlikely (not impossible) for a lien to be put on the property in the situation you describe. But, that does not stop a lawsuit over a claim by a sub.
Your response here is very typical, you want to tell us all of the facts that make it "unfair" for a sub to sue you. Take our advice, the sub can sue, and if you don't retain counsel once you are served in the lawsuit a default judgment can be obtained by the sub.
So: 1) demand a release of liens from the contractor for all subs and suppliers; 2) hire an attorney if you are served with a suit or lien.