Internet Exchange or Carrier Hotel: Let's take it as a given that the Internet is a network of networks. People and companies running applications connect to the Internet using a variety of service providers (Level 3 Communication, Comcast, Sprint, Verizon). All these networks have to, at some point, talk to each other.
If I want to watch something on Netflix and I'm connected to the Internet via Time Warner, but Netflix is connected to the Internet via Cogent Communications, my HTTP request for netflix.com has to leave Time Warner's network, travel through Cogent's network, and then come back to me via those networks. This happens at an Internet exchange, or IX. Basically, they're buildings where cables and routers connect other networks to each other. Sometimes, they're called carrier hotels, because they're where all the “common carriers,” as it were, “check in” with each other (come for the network infrastructure, stay for the dad-joke terminology).
Internet exchanges are scattered all over the place (TeleGeography maintains a pretty cool map of them), but a lot of them end up in areas where there's a lot of “Internet backbone,” another vocabulary word that infrastructure enthusiasts use but never really explain particularly well. Basically, “Internet backbone” is a word for a really large concentration of network infrastructure (i.e., fiber-optic cables) converging in a particular area due to various political, historical, and environmental conditions. Lots of Internet traffic moves through these areas. It might be more accurate to call Internet backbone “Internet spinal cord” or even “Internet spinal fluid” for biological-metaphor accuracy, but that sounds kind of gross.
There will be a story about a pretty cool Internet exchange later on in this series, but there is a bit more America to travel through first.
Colocation Data Center: Data centers are not in and of themselves cloud infrastructure, and data centers have been a thing long, long before people excitedly talked about The Cloud. The premise of colocation is kind of what it sounds like: Companies put their servers in the same place. In general, co-located servers are hardware that individual companies bring into a data center—they own the equipment, and they put it in a particular data center.
One distinguishing trait of cloud infrastructure compared to vanilla co-location is partly that data doesn't really live on a single server. Databases are distributed across multiple servers, stored in fragments sometimes called “shards” (a term that apocryphal rumor I desperately hope is true attributes to the MMO Ultima Online). If you are using cloud infrastructure and you're not a giant company like Amazon, you usually don't own any of the hardware—it's more like you're renting it. If, in your life, a Man from Sales named Chad (he is, always and forever, named Chad) ever tells you about “platform as a service,” this is literally all he means.