When you select a Minecraft server you put its code into Minecraft and assuming the server is currently running and accepting new players you will 'join' it.
You will then appear in the server spawn as if you were playing a single player world. Feel free to pick a direction to walk in and get started playing.
Anyone currently connected to that server can see you and interact with you and what you build. They might be a *very* long way away from you however, and the server does not necessarily tell them where you are, so you could be on a low-population server a long time before you find anyone.
What you do in the server is persistent, it sticks around, but be aware that many servers occasionally reset themselves, reverting *everything* and everyone back to start.
As for the choice of server, that's really up to you. I would recommend any server that has anti-griffing rules in place for a start, otherwise you will run into someone whose only goal in Minecraft is to ruin everyone else's fun.
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Mods are add-ons to the PC version of Minecraft that changes something about the game. There are hundreds, maybe thousands, of mods, so there's no easy way to describe them.
A server can host a modded version of Minecraft. If it does you will need those mods before you can enter the server, though I believe some servers auto-download the mods for you.