"Root" is the common way to refer to the fact to the user account which has superuser rights on the device. Superuser rights imply full control over your phone's capacities, which are determined by the software and the hardware of the device. Non-superuser accounts have limitations, usually established for the sake of software security (for example, not allowing a user to disable critical functions such as phone or GPS), hardware safety (not allowing to change the processor's frequency speed, aka overclocking the CPU) or, in some cases, for someone's profit (disabling wi-fi tethering is a typical one as users are forced to buy data plans in order to use their devices to tether). In Android's case, Superuser is composed of an app* and a set of instructions to give this app root rights.
The bootloader is a program which accesses the internal storage and, basically, controls the loading of the programs and data necessary to initiate the sequence of launching the OS (aka booting, thus the name bootloader). We say that a bootloader is locked when it is programmed in such a way that it won't allow to load the device if it detects data from third-party sources. Unlocked bootloaders do allow these data and will load what these data contain, which will be usually a modified version of the phone's firmware, known as "Custom ROM".
As we can see, there was the fear that one would need to have unlocked bootloaders (i.e the ability to load third-party data in the booting sequence) in order to gain root rights. This would have been terribly inconvenient because only a minority of users will have unlocked bootloaders, since the phones with unlocked bootloaders by default are the ones sold carrier-unlocked. Said carrier-free phones will be a minority as the phone is pretty expensive: most Xperia Plays will be sold using carrier subsidies.
However, our fellow member Chainfire found an exploit (a programming loophole) in Gingerbread which allows said app and instructions to be loaded without the need of touching the bootloaders. Basically, his Gingerbreak program allows the user to have root rights without loading any bootloader-sensitive data. This widens the possibilities for the majority of users who will buy this phone with a subsidy.
*edit 3* Later on, it has become possible to unlock the bootloader even for SIM-locked devices, thanks to different unlockers like Alejandrissimo and Jinx13. This means that, provided that you pay what they charge, you can have your device set up without the bothers of having to buy a SIM-free phone. What's better: their method of unlocking the bootloader SIM-unlocks the phone as well.
I think this is more or less a succint framework of ideas which allows us to define "root" and "unlocked bootloaders" properly. As they say, knowing is half the battle, so I hope this nugget of information allows users to deal with their phones with more confidence.
*edit* as of version 2.3.3, this information has changed. Gingerbreak doesn't work with 2.3.3 anymore, and it seems now that the only way to root with locked bootloaders is to flash a 2.3.2 rom with FlashTool and update through OTA.
*edit2* note that this explanations pertains mostly to the Xperia Play. Other devices are NAND-locked, which means for us that they need unlocked bootloaders to even get root access.
*Note that Android is built in such a way that every app is a user account with limited rights (the ones specified before you download it). Thus, Superuser is a user account as any other app and the set of instructions is meant to give this user account root rights.