This is a different beast than e-mail (which generally involves a few delays), instant messaging (IM) works by directly routing a message through to a recipient while they're on-line. This allows you to chat almost in real time—passing little notes back and forth between yourselves.
There's quite a few IM systems available, the most widely used ones were Jabber, ICQ, MSN messenger, and Yahoo messenger (out of which only Jabber could communicate with the other systems, they all deliberately only work within themselves—yet another example of corporations trying to monopolise markets).
Many of them don't exist any more, that's what happens with proprietary systems if they can't keep their customers hooked, as opposed to an open standard that can be used by anybody capable of programming for it (such as email, which some people still keep trying to kill off in favour of their propietary messaging system). Now we're left with personal messaging on phones (which really took off once phone companies stopped charging per message), Facebook messenger, What'sApp, Zoom, Skype (soon to die) and few other proprietary messaging systems (all of which will face the ultimate fate of proprietary systems).
You can still use some of them by getting yourself an IM client program (Gaim's a good, and simple, one that can actually use all of those systems within the one program), then you sign up for an account on one (or more) of the IM systems, look for people to chat to (some of them have search systems to find people with certain interests, or to find specific people), let people find you, or tell all your friends your username on the system (just like telling friends your phone number).
You can find my IM details by searching for them (I use my real name on them, though many don't), or you can simply get them from my contact details page.
IM systems generally provide at least two basic features: The exchange of messages between people, and showing who's currently connected and available to chat (you build up “buddy lists”, and your list shows their statuses).
And some offer additional features, like being able to store messages for off-line people (who'll receive them when they get around to connecting, much the same as e-mail), the ability to exchange files, and to set up connections for other things (like group chats, playing games, voice conversations, etc.).