I was wondering if it's possible to host a Minecraft Spigot server on a website?
It is absolutely possible to host a Spigot server on any server that you have full access to, but not on any public "web hosting" service.
If you have a "server" on which you can run and download applications, you can just download the latest server jar here, and start it with the following command:
java -Xms1024M -Xmx2048M -jar minecraft_server.jar nogui
You can omit the
nogui part if you want to see a visual representation of your CPU usage.
For plugins, you can put the zip files into the plugins folder (same as local).
After you start the server, it will be accesible from both the default IP address of the machine (if you have multiple), and all domains that are connected to this IP.
I'm going to have to make some assumptions based off information provided.
No. You cannot embed Minecraft to play on a webpage. You'll need the game client to run it.
If you want to host the Minecraft server so it's available from locations that aren't your home computer you'll be far better off paying a hosting service for a server with few slots. Since you've indicated you want to play alone this should be cheap, likely less than $25 annually (~$2 a month).
Using an server hosting service would also allow you to whitelist players from the game and prohibit unauthorized players from joining. Even if you don't using a hosting service's ability to whitelist the odds of any other players guessing your server IP address is so low it's practically nonexistent. I wouldn't worry about that.