LogMeIn Free is gone, but don't panic: You can find alternative remote-access tools that cost the same low price of nothing at all. Whether you need to access a document, collaborate with a colleague, or support several PCs, try one of these free tools to get back into the game.
I've been using TeamViewer for years to help out family and friends, and it has always been reliable. Simply download the program from the company's website, and then install it (or run it without installation, if you desire) on both of the PCs you want to connect. During installation, you can also set the program for unattended control.
For ad hoc use, simply run the program and log in from the controlling computer. The two components will connect, and up will pop a window containing the desktop of the computer to be controlled. TeamViewer installs as both a server and a client, so you can use it to take control or to allow control.
TeamViewer 9's cooler features include the ability to open multiple remote sessions in tabs (as in a browser), cut and paste between computers via the clipboard, and drag and drop files from your desktop to the remote desktop. It's a mature, stable, practical tool for anyone's remote-control needs. Note that you'll get the occasional message about upgrading to the pay version if you use TeamViewer regularly to connect to a lot of different PCs. You're on your honor for that one.
Windows Remote Desktop
Although Windows Remote Desktop doesn't support true screen-sharing (the screen of the controlled computer goes black instead of staying live) the way services such as Join.me and TeamViewer do, this built-in tool is free and fast, and it allows complete remote control over PCs. There's even Microsoft Remote Desktop for the Mac, so you can remotely access your more artistic acquaintances' Apple products.
The basic concept behind Windows Remote Desktop is to let users control their office computer remotely so that they can work from home. Hence, although all versions of Windows (Basic, Home, and so on) can establish a Remote Desktop connection and control a PC, only the Professional, Business, and Ultimate versions of Windows can be controlled.
As most office computers are one among many on a network, you need to have the office router tweaked to forward a port (3389) to the PC you want to control. You can edit the Registry to allow control of more than one PC by adding more ports, but that's a very techie task.
Windows Remote Desktop works great once you've set it up, but if you want to control multiple PCs on a regular basis, the next option might be better for you.
VNC, or Virtual Network Computing, isn't itself a product, but an open-source remote-control and display technology that's implemented by Tight VNC (free), Ultra VNC (free) and RealVNC (free and pay), among other parties. VNC isn't hard to use, but it's not as simple as Join.me and TeamViewer, which don't require user knowledge of IP addresses.