There are a few ways to join a Discord server. Method number one is of course to make one yourself. But you probably care more about joining an existing one. Someone already in that server can invite you. If you have a message history with them, they can right-click your name, click Invite To Server, then select the server to invite you to. This only works if they have permissions to invite you to that server, of course.
Instant invites are the way to go, though. An instant invite is created when someone in a server creates an invite. These links can be customized to last a certain amount of time, or to only last a certain amount of invites. Most publicly-posted invites will be set to never expire, and have infinite invites. This is ideal for listing sites, like Discord Me, or for social media links. Again, not every server member has permissions to make an instant invite link.
Discord also features a Server Discovery section, which shows official servers for games and some communities.