Spoiler alert: Apple Remote Desktop and Windows Remote Desktop are not included. These are great for enterprise solutions, or for people who enjoy running VNC servers in their houses. For most folks this is not only overkill, but also puts a burden on a less technical person to be able to use those tools. And if you aren't actively maintaining that remote network, it's likely that port forwarding and such can get obliterated while you're gone, rendering those tools useless.
First up, there's what you already have installed on your computer: Messages. Messages will allow you to text with the person remotely either on an iOS device or a Mac, send a screenshot, and even screen share. Options for this are found under the 'Details' button in each individual chat. Keep in mind this only works Mac to Mac and both must be running Yosemite, so if your relatives have yet to see the light this isn't the option for you.
Behold the majesty of screensharing in Messages!
Skype is a nice cross-platform common option, since lots of people already have it installed. You can use voice or video chat, and screenshare with the other person, even controlling that other screen remotely instead of walking someone through doing the clicking for you. Another reason Skype can be handy is that in the worst case, usually there's another device around that can be used for Skype and that camera can be pointed at the thing having trouble (yes, I have actually resorted to this).
Skype in screensharing mode. Find this under the Conversations menu.
Another good cross-platform screensharing option is TeamViewer. It's solid, and even offers free apps for screensharing on iOS and Android devices, both to view a mobile screen and also to view that screen from a mobile device.
TeamViewer's Mac app. Just read off the numbers and you're all set.
A quick option, especially if you don't get the chance to lay groundwork ahead of time, is Join.Me, a lightweight screensharing app by the LogMeIn folks. It's designed to convince you to pay for LogMeIn, but if you only have an occasional need this will do the trick. Go to a web page and after a quick download you're on your way. This doesn't include audio, so you'll need to keep the other person on the phone to walk them through the troubleshooting which might be awkward for people who don't regularly use their computer while on the phone.
You don't need to set anything up in advance to use Join.Me. It's right in your browser.
Another option that looks interesting is Chrome Remote Desktop, particularly if neither party is using the default browser that came with their OS. I haven't tried this, but I know people who have tried it and generally found it useful.
Hopefully with the holidays upon us, you will get a chance to set people up properly with a remote support tool in case you need it in the future. And as long as you can establish that anything which sounds a little odd should be run by you first, this can save endless hours on the phone in the coming year, even eliminating the dreaded phone call about how Microsoft called and said they had a virus…