The internet as we know it today, puts the information highway at your fingertips. Instant access to mountains of information and entertainment. You use it on a daily basis, and it’s become so interwoven with the fabric of your identity that you don’t even think about how the hell it works. But when it does stop working, you instantly feel like the apocalypse is imminent.
At a very high level, the internet can be thought of as a network of networks (that means, it connects tons and tons of small/large networks together). At the very top there are a select number of top level global network connectivity providers, mostly telecommunication companies (Think large phone companies). Who Interconnect (peer) their networks at various exchanges all over the world. These top level companies, own the physical infrastructure that we seldom don’t notice in our day to day lives (Telephone wiring across highways, large underground wiring etc). Seeing as they literally sell connectivity, at that top level the providers generally peer as equals, as it is in their interest to do so (they’re own network becomes far more valuable if it can connect you to the other networks as well). That’s literally what makes the internet The Internet.
There are also other network providers and ISPs (Internet Service Providers) lower down on the totem pole that either have somewhat less equal peering agreements (say, paying a small fee), or just outright purchase connectivity from the top providers. These guys generally have smaller networks and so have less “connectivity” to offer. Basically, where you are on the totem pole is determined by how valuable it is to peer with you, which is usually directly related to the size of your network and what it is connected to.
Now, if you mean physically what/where they are connecting, that would be at an exchange point somewhere. Here is a handy map I found of internet exchange points around the world. These are the actual locations, generally large network infrastructure installations most people would just refer to as “data-centers”, where the various network providers and ISPs actually physically run their fiber lines to. The lines are usually buried along roads and railways and such, though they may be on poles sometimes too. In order to interconnect at these exchange points, they rent one of several physical spaces in the building(s), install routing equipment, and run their fiber lines to it. They then run lines between their routing equipment and the routing equipment of the various other networks they have agreements with.
the entire architecture of the internet is set up in such a way that there are peering points which create multiple paths to places. If you lose a peering point, traffic will re-route. If you lose enough of them, or if there aren’t many of them in the first place, you can theoretically not find a path to certain parts of the internet. For example, the closest peering point to me is MAE East (“metropolitan area exchange”). I would guess that 70% of everything I do on the internet goes through MAE East, and if it went down, I would still be able to do those things – but it would be slower, and through a longer path. This is what happens, when undersea cables get cut, and suddenly a remote part of the world that is only serviced by a small number of connections has vastly reduced connectivity, or falls off the internet entirely. Take a look at this map – Madagascar and Mogadishu probably have 90%+ of their internet connectivity that leaves their countries going through one of two cables that each carry just over 1Tbps – and probably land at the same place. That in short is the internet as we know it, organized chaos. For the time being, this model works but there is always room for improvement!