Root: Rooting an Android phone is simply adding a file to the system that allows other apps to elevate their permissions and read, write, and execute anything on your device. In this case, anything means anything -- if it is user editable or actionable, you can do it with root. This is both powerful and dangerous, so be sure to get all the answers and be clear on the subject before you do it.
SIM or Network Unlocking: This allows a phone bought for use on a particular network to be used on another network. If you buy a phone designed to only work on Orange (or AT&T for an American example), to use it on any other network, you will need to unlock the SIM programming. It's done without rooting or modifying any firmware in your phone or tablet.
Remember, the networks have to be compatible. A phone with radios designed for one carrier may not provide 3G or 4G service on another, and sometimes they won't work at all.
Bootloader Unlocking: All Android devices ship with a locked bootloader. Some are very easy to unlock, like Nexus devices, some need a little hacking to unlock (like most Samsung devices), and some are encrypted and designed to be very difficult to unlock (hello, Moto). Bootloader unlocking allows you to flash (write to your phone's "hard drive") image files that haven't been signed with the official key from the folks who made your phone or tablet. A locked bootloader can flash a new recovery provided in an OTA update because the recovery was signed with the right key. It will fail to flash a custom recovery like ClockWorkMod. An unlocked bootloader will flash anything that fits, as long as you tell it to. Once a custom recovery (or sometimes a "Super" boot image) is flashed, you can install and erase custom built system firmware at will. Again, this means you need to do your homework before you start fiddling with things.