What computers my family should use
TL;DR: Family should buy Chrome OS devices (e.g. Chromebooks) unless they have special needs which aren't supported on that platform, in which case they should get a single Mac and Chrome OS devices for everything else.
[last updated: 2014-02-02]
Chrome OS/Chromebooks for cheap and simple
Chrome OS is Google's operating system for the web. You can think of Chrome OS devices (e.g. Chromebooks and Chromeboxes) as computers that have nothing more than a Chrome browser on it. If you think about it quickly this might sound rather limiting and a silly idea, but if you stop to think about the idea of a computer that is nothing more than Chrome you will come to realize that it's a great idea when it can meet your needs.
First off, Chrome OS devices are simple. Think about what you currently use your computer for. Do you ever do anything that doesn't involve your browser (or couldn't be done in the browser; check the Chrome Web Store if you think there is something you can't do)? Chances are no, and so having a computer geared entirely towards running a browser makes sense.
Chrome OS, because it's a browser by Google, stores everything in the cloud. This means you do not ever need to worry about losing any data, having to copy over settings, etc. For instance, if you want to wipe your Chromebook and reset it to its factory settings it is not at all scary since all of your data is stored in the cloud at Google and will simply be set again when you log back into your computer (I have done this without worry and it's rather refreshing).
These devices are also eminently sharable. Since you are literally logging into your Google account it means everyone in the house can log in and have separate accounts and settings; no more bringing up the computer and finding someone has logged you out of Gmail. It also means people visiting you can just use your Chrome OS devices and not lug their own computers with them.
Because Chrome OS runs nothing but Chrome is means these machines are extremely secure. There is no such thing as a virus for these machines; they are as secure as Chrome is. And since the software is updated every 6 weeks they are constantly improving in speed, security, etc.
Finally, the machines are cheap; under $300 cheap. Since they only need to power Chrome and not worry about running any random program that is thrown at them they can avoid wasting money on unnecessary performance-driven parts. This also has a side-benefit of often times having no moving parts because they don't need a fan or spinning hard drive, making them very reliable. They also don't have a ton of flare so they are understated.
You can get either a Chromebox or a Chromebook. A Chromebox is a desktop computer, but they are extremely small. You hook up a keyboard, mouse, and monitor and you have your keyboard. My father-in-law has one and likes it and finds that it meets his needs.
A Chromebook is a laptop. Typically about 11", they have good battery life and are extremely light. Google typically has two Chromebooks that they are pushing: an entry-level one and a high-end one. They are designed by Google and thus work with Chrome OS well and with no fluff. Andrea loves hers for its lightness, battery life, and having a keyboard (in comparison to a tablet). She actually won't use any other computer in the house but her Chromebook.
Buy whatever the latest Chromebook is.
Mac if you need more than "simple"
Sometimes Chrome OS can't meet your needs because you need to be able to run a specific program that is not available on your phone or in the browser (whether it is a website or a Chrome app from the Web Store). In those instances you will need a more traditional computer. Since Chromebooks are so cheap you should only need a single Mac for the occasional use and everyone else can share or have their own Chromebook.
For laptops you will want a Macbook Air. They are light, fast enough, and Apple's cheapest laptop.
If you want a desktop solution you have two options. The first option is to buy an iMac. That gives you a computer and monitor as a single unit which makes the decision process dead-simple, but it also means that when it comes time to upgrade the computer you also have to pay for a brand-new monitor. These machines start at $1,300 for a 21" version and $1,800 for 27".
The second option is to buy a Mac mini and a separate monitor. This gives you the ability to upgrade the computer and the monitor separately. The Mac mini starts at $600 and a good Dell monitor at 24" is $370; 27" is $700 from Dell, $1000 from Apple . That means this option compared to the iMac is $330 and $500 cheaper than the comparably-sized iMac; knock off $400 from the savings if you want a Mac mini of comparable performance.
Buy the Macbook Air. If you want a desktop then go with the Mac mini and a separate monitor so you can upgrade the computer and just keep using your monitor (buy Dell if you want to save money, Apple if you want a pretty monitor).
Linux if you're daring
Since family members ask about Linux on occasion, I will say it falls in between Chrome OS and Mac. While Linux can run apps, the apps that would necessitate having a Mac probably won't have a comparable Linux app since it will only be available on Mac and Windows. But Linux is free and can run on older hardware, so if you have an old Windows computer laying around you can try putting Linux on it and see if it is usable for your day-to-day needs.