Okay, it's time to tell new users a secret about Mastodon.
Mastodon is (whispers) part of a much bigger network 😮
Mastodon servers use an open standard called ActivityPub to talk to each other. That's how you interact with people on other Mastodon servers.
But, dozens of other federated networks also use ActivityPub, so (this is the good bit!) you can follow people on these other networks from your Mastodon account!
For example:
PixelFed (https://pixelfed.org) is a photo sharing network, here's a random example account @Iancylkowski
PeerTube (https://joinpeertube.org) is a video sharing network, with a p2p system that allows videos to go viral even on small servers, here's a random account @craftykat
BookWyrm (https://joinbookwyrm.com) is a social reading site, an open alternative to Amazon's Goodreads, here's a random account @mouse
Together, Mastodon and these other services form... The Fediverse 
(whispers) Click on the link...
https://framatube.org/w/4294a720-f263-4ea4-9392-cf9cea4d5277
The link you just clicked on leads to a video on PeerTube, on a server called framatube.org.
BUT... if you copy and paste the same address of the video into the search box in Mastodon, and search for that link within Mastodon, it will appear to you as a Mastodon post 
You'll then be able to watch it (of course), but also you'll be able to comment on the video by replying to the post within Mastodon, like the video by favouriting it, and follow its creator by following the account that posted it.
This is the Fediverse in action, totally different services interacting through a common open standard.
It can be hard to get your head round this at first, but it's incredibly useful and once you get used to it simply how things should be.
Hi, one of the influx on new users here.
From a user interface point of view this is one thing it would be cool to iron out, if possible.
When I go to another server that's not my own, it would be nicer to just sign in there (using my existing account as a "single sign on" type thing) and being able to write my comment without having to come back to my own server first.
I know there are probably technical reasons for why that has not been implemented but I think it would be cool.
I'm not completely sure what you mean by "going" to another server?
If you follow or interact with someone within Mastodon, you will see their posts in your timeline and you can interact with them there without going anywhere else. This is true whether the person is on Mastodon or some other Fediverse network.
If you can't find them on your server, you can put their address in the search box and it will make them appear. The same goes for posts (which is what I mentioned in the original post).
I might have misunderstood what you mean though?
I think i have explained better here
No, i don't mean server, exactly. I mean a different instances web client, more precisely.
Ah, ok, I see!
Right, okay...
You don't have to go to their server and click follow.
It's much, much easier just to copy the URL of their profile into the search box on your server, click search and interact with it from within your own server.
It means no signing in, no extra work, it becomes just like any other person you see in the timeline.
Same goes for copying and pasting URLs of posts.
Ok, yes. But still a little klunky i think.
A different solution would be instead of having to copy paste the url...
1/ you only have to click on a link to someone elses profile
2/ your web client recognises this is a link to another mastadon client
3/ and then the profile is automatically loaded in the your web client instead of switching to the other server's client.
Yeah, that would be nice! Maybe if there was a certain formatting in a Masto URL it could recognise it as a Masto server and launch a Mastodon app?
If you're on github, do you want to suggest it on the Mastodon github? I could do it if not.
