Yes.
Unspent means that they did not spend it yet. They received it, and they have the bitcoins. They did not send it to anybody else yet. It is still where you sent it.
When it changes to "spent", that will mean that the person that you sent it to no longer has it. They have "spent" (or sent) it somewhere else.
If they want to use the bitcoins which they received from you for a product or service, then they will need to create and send a transaction to whomever is supplying them with that product or service. When they do, the bitcoins that they received from you will be "spent", and the person that they send them to will receive "unspent" bitcoins that will wait for that third person to use them.
Alright, now I got it! Thanks for your explanation