I will break down how things used to work and slowly come back what serverless means as on date.
Say I want to host an application. Here is what I used to earlier -
Case 1: Buy a physical server. Setup the hardware, the software the networking, the storage, and all other dependencies and then host my website. Problem with this, too expensive to buy an actual server. Difficult to upgrade in order to scale. No-fault tolerance or load balancing etc.
Case 2: Rent a server from the many hosting providers out there. Buy a shared space or a dedicated one. Put my software on it. Get charged for using it or not using it as it's on rent. You pay X amount regardless of usage.
Case 3: Host my application on the cloud. Get charged for what I use and for how many hours I use. Say my application wasn't used or I switched my database off for some reason. I won't get charged for it. Say I want to scale my RAM to adjust to my user visits, I can do that with a single click. Get more instances of my CPU. You guessed it, can do it with another click(s) or so.
So serverless basically means, pay-per-use model. You pay for the amount of time you are using.