Mastodon, and Lemmy are different pieces of software, that communicate using a mutually-compatible protocol.

Sort of like how you can make phone calls with an Android and an iPhone, even though they are very different.

Lemmy.world is a Lemmy instance, like Mastodon.social is a Mastodon server. It is basically a server that runs the Lemmy software, and because Mastodon and Lemmy use ActivityPub, the mutually-compatible protocol, they can communicate.

In relation to lemmy.world specifically, Ruud, one of the operators for lemmy.world, also runs mastodon.world(?), one of the major Mastodon servers.

lemmy.world is a lemmy instance. The admins also run a mastodon instance.

Mastodon is a twitter-alike software that can be deployed on different servers. So everybody who wants to, and has a shiny domain, can run said software…

Lemmy is a reddit-alike software that can be deployed on different servers. So everybody who wants to, and has a shiny domain, can run said software…

Often they choose subdomains like lemmy.example.com and mastodon.example.com . However, lemmy and mastodon share a same communications protocol. Thats why they are in parts compatible…

You can basically be on your mastodon server and then some lemmy content shows up.

The operator of mastodon.world choose to share a lemmy installation via lemmy.world.

Both Lemmy and Mastodon are fediverse software. So while the uses are a little different, you can use both to interface with the same content.

Mastodon doesn't have a concept of communities, it creates users for communities instead.
On Mastodon the custom is to follow hashtags and users.

What happens when you write a post here in the scope of Mastodon:
All Mastodon servers that are federated with lemmy.world (these can defederate instances same as Beehaw defederated Lemmy.world) will mark that your user (@Blamemeta@lemmy.world) made a post and the user created for the community will boost it.
An answer made to a post on a Mastodon instance (that is not defederated from Lemmy.world) will eventually be synced to originating Lemmy instance.

Since we don't have a custom of using hashtags on Lemmy, some content is probably getting unseen in Mastodon. But if we started using these, it should start appearing.

But in the end both are separate things, like reddit and twitter are. Separate logins, separate code, etc. It's only the ActivityPub that makes the content cross.

Random example