It could be several things, such as apps updating, surfing the internet, streaming music/videos. Are you encouraging them to use wi-fi as much as possible? You may want to check into Usage Controls (costs $4.99 per month):
Yes, you can use alot of data very fast which is what I found out AFTER I upgraded and lost my unlimited data plan. If you do alot of web browsing, downloading pictures and video, watching Netflix or YouTube alot that will eat alot of data. Also the android operating system has a lot of apps that automatically update, every time your weather updates or Gmail or Twitter or Facebook updates(just to name a few) you use data. you can go into settings and see what app is using alot of data and you can download several apps from the Play store which will tell you what is using data. Always stay on WiFi as much as possible. Hopefully this helps
There are so many things which can eat up data. Several of them have already been named. Streaming music/videos, watching movies, apps upgrades, messages/email eat up some data. I'm not sure about texting, but that might. As far as automatically upgrading apps, you can shut that off, depending upon your phone. Tikibart suggested Usage Control, an excellant idea. The $4.99 per month may save you more than that per month. Data is expensive.
Text messages don't use up data (they come from a different part of your plan, and they use the phone, not the 3G or 4G data connection, which is why you can text when you can't surf the net). Also taking pictures won't use data, though uploading and downloading photos, videos, and music will. I'm not sure that updating an app uses your data allotment, but any kind of e-mail does, as of course does all Web browsing. There are some apps that access the Internet without using the browser, and that will use data; I think the Accuweather widget is one of those.
To 6109: the Web browser on the phone should have a history that will show which pages have been accessed, but probably not how much data it used to access them.
It'd be kind of a pain, but you could check My Verizon every day and keep track of the data usage by the particular phone in question. When you see that an unusual amount has been used on a given day, ask the person what they've been doing that day. Maybe they watched a movie or sporting event online, for instance.