Huh, interesting. There might be a call for a cell phone glossary somewhere on the Web.
So basically, a cell phone is a "cellular phone." The name "cellular" comes from the way they work. A portable phone is basically a radio transmitter and receiver. It communicates with radio towers that are connected to the phone network. Each radio tower has a range of anywhere from a quarter mile to a couple of miles. The area served by one radio tower is called a "cell."
The magic that made cell phones possible was in figuring out how to get the towers to automatically hand off a signal from one cell to another as the phone moves around. This is harder than it seems, because if you're talking on the phone when you move from one tower to another tower, the first tower has to tell the second tower who you're talking to, and the second tower has to establish a connection with the person you're talking to using its own telephone line seamlessly and without dropping the call. There's a lot of very complex behind-the-scenes magic to make this work.
The difference between a cell phone and a smartphone is that a cell phone is just a phone. It makes and receives phone calls; that's it. A smartphone is basically a computer. (In fact, there's no "basically" about it; it is a computer, roughly the equivalent in terms of power to a 2006-era iMac.)
A smartphone has three radios: a cell phone radio that makes voice calls; a digital radio that offers an always-on, permanent connection to the Internet through the telephone company's network; and an ordinary WiFi module that connects to WiFi the same way a computer does. Your cell phone company charges you separately for the voice connection and the Internet connection, just like your home has separate billing for an old-fashioned land line and cable Internet.
"Roaming" is what happens when you leave an area where the phone company you buy service from has towers. For instance, T-Mobile builds cell phone radio towers, for the most part, only in cities. They don't have a lot of radio towers in the countryside. AT&T and Verizon both have a lot more radio towers outside cities, so when you leave a major metropolitan area and you're a T-Mobile customer, your phone may start talking to radio towers owned by someone else, like AT&T. Similarly, if you travel from the US to Canada, you'll be on a different network.
Many, but not all, networks charge you more for using someone else's radio towers. Those are "roaming charges." For voice calls, they're quite low. For Internet connections, they can be very, very expensive.
Apps are programs, exactly like programs you run on a home computer. Often, they're the same (I use Pages for word processing on my computer and I also have Pages on my phone, which means I can do word processing on my phone.) Apple provides a service that allows you to work on your files on your home computer and also on your iPhone or iPad, and switch between them. I can edit a word processing file on my computer, hit Save, take out my iPhone, run Pages, and presto! There's the word processor file I just saved from my desktop.
They also run apps like Web browsers (I run Safari on my iPhone, and it works basically the same way as Safari on my Mac.)
