Well.. hopefully this will make sense, but Plex needs three key parts to be active at all times to function properly.
Plex Media Server (PMS) - This is the backbone of Plex, and it needs to be running live in order for anything to be watched. It can run on a computer such as your laptop, or it can also run on what's called a Network Attached Storage Device (NAS; see below). There are a few other options that people are fond of which may get mentioned in additional comments.
Your Files - Once you have PMS up and running, it will need to be able to access your files. You can store these files on the same unit or computer that runs PMS, or you can buy a NAS device to house them. Some people prefer to purchase a NAS that can run Plex. Most can, but you'll need a higher end one if your files aren't compatible with your Plex Client. You really only need a NAS if you've got a LOT of files, otherwise you can probably get away just a PC with a decent sized hard drive.
Plex Client - This is the app or piece of software you're using to play back the files on your chosen device. Lots of Smart TVs or devices such as a Roku have Plex Clients available for them. There's even an app/program for the PC known as Plex Media Player (PMP).
In theory, you could run all of these pieces from a single computer and be fine, or you can spread them out across multiple devices and it can work well that way too. The key, is that all of them need to be available when you want to watch your files. If one of these components is off, or in an unwakable sleep state, then you're not going to get very far.