I am sorry if I am saying something wrong.
“Relay server” is a very generic term. What’s the context?
Usually VPN refers to service/system that can create secure tunnel between to networks. Relay server in gamedev context usually refers to system where multiplaye game runs practically as P2P but game clients don’t talk directly between them and messages are routed using server hosted by developer. VPNs technically are traffic relay servers in more generic use of the term.
VPN is a virtual private network. It makes remote computers feel like they are a part of local network when they talk to each other. One common use is to bring remote company locations into a single network. It can also be used to pretend one remote computer is in a different location by joining a VPN that “exits” to the public internet elsewhere.
A relay server is a server that takes messages from one computer and sends them to another. A central computer that coordinates all remote computers in a VPN is one type of relay server. But relay servers can be used for many other purposes outside of VPN. You can relay any other type of traffic, not just VPN.
A relay server typically only forwards a single specific protocol (eg email, gameserver shard, https proxy, etc), while a VPN forwards anything/everything.
Some people are really mean about teaching something. I was searching for the same question and found your post. I dont know that much about vpns and relay servers and when i look it up i encountered unrelated information in the web. So you asked here what you should know about relays and vpn and sompe people started “Achtually a rly srv a mail srv and you are so ignorant and dont cnow anything A-hyuck 
”.
Relay serverhttps://askleo.askleomedia.com/wp-content/uploads/2009/10/email_with_relay_server.png
VPN http://mspoweruser.com/wp-content/uploads/2010/02/hamachi_thumb.jpg
A relay server is a server that relays email from one machine to another machine. Most email uses at least one relay server. The relay server will hold on to an email and retry later if the destination is unavailable.
This also has nothing to do with games.
Those are both very bad images. What does this have to do with gamedev?
Why, if I point out, do you just give a low rating to me without refuting?
If there is something wrong, please tell me
A relay server is a server that relays email from one machine to another machine. Most email uses at least one relay server. The relay server will hold on to an email and retry later if the destination is unavailable.
This also has nothing to do with games.
You can really say that the relay server has nothing to do with the game, right?
You can really say that the relay server is used only for mail, right? 
some game actually using a relay server in a multiplayer game. What is this?
If you connect to the same VPN with the server and client model and play a multiplayer game, this VPN will effectively become a relay server.However, this is what is often used in games (steamworks, hamachi).
(Perhaps even if you connect with p2p, the VPN will be a relay server.)
In the first place, even when connecting to Google using a VPN, that VPN can be said to be a relay server.
If this is wrong, Refute this instead of pushing a low rating
I did not give you a low rating.
You posted a picture of an email relay server and asked what a relay server is. I told you that an email relay server holds on to email and delivers it to another server. It is about email, not games.
If you are not talking about email, you need to provide context.
You posted a picture of an email relay server and asked what a relay server is. I told you that an email relay server holds on to email and delivers it to another server. It is about email, not games.
I’ve never heard such a stupid excuseYou said “A relay server is a server that relays email from one machine to another machine”, right? (This is by definition a mistake)
And you yourself said that the relay server has nothing to do with the game, right?
“what a relay server is”
This isn’t what I’ve heard, it’s written very large on the top of the post, “What’s the difference between a VPN and a relay server?”
You say you need context, but the person answering this question in this community says that the game and the relay server are not related.
I don’t see how someone who makes such a statement needs context.
Also, I’ve already left one more comment
Look at that.
If you want to find out that a relay server is used in your game very easily, check out “Unity relay server” with google
Learn to ask better questions if you want better answers.
Do you admit that what you were saying was wrong?
“Learn to ask better questions if you want better answers.”
What I want to know in a conversation with you is whether you were wrong (although what I wanted to know in this post was different).
I’ve known you for like two minutes of my life and I just don’t care enough to figure out what you’re asking, and you don’t care enough about the answer to ask a clear question. I’m not gonna decode your rambling and you’re gonna have to fine someone else to answer whatever it is that you’re asking.
I wonder why you don’t admit mistakes so much.
I also want to know if I was wrong, so I wanted to know this answer
It may not make much sense,
What I find most annoying is that you don’t explicitly say that this is wrong, even though there is a case that some people who come here to post will think this is right.
Also If you need context, go back to the comments
I have already written
Why do you wonder about me? I don’t wonder about you. You are a complete stranger on the internet.