What sort of features does this typically include?
Typically, any camera with wi-fi capability can basically be a wifi hotspot, so you can use an ad hoc network between the camera and a smart phone or tablet to transfer photos from the camera to the device, use the device's GPS positioning to geotag photos, or to establish a remote control path to the camera's basic functions. A mobile app on the smart device handles the communication.
Does this mean if I input my username and password into the camera, it will automatically upload pictures to Google+ when I take them?
Not exactly, it's a bit more of a pain. You have to turn on the wi-fi in the camera (continual use would run down the battery), then connect to the camera's network on a smart device. From there, you can transfer images to the smart device, and use a mobile app to upload the pictures to your social media of choice.
A special case in wi-fi features that I use is the one that Fuji's building into its cameras to send pictures directly to its wi-fi smartphone instant film printer, the Instax SP-1 from the camera menus. However, that only works with my Fuji X100T. With my Panasonic GX-7 or Eye-Fi card, I have to do a wi-fi/app dance as follows: connect iPhone to GX-7/Eye-Fi card, use Panasonic/Eye-Fi app to transfer photo to camera roll on iPhone, switch wi-fi network to Instax Printer, open Fuji Instax Share app, print photo from camera roll.
Or are the Wi-Fi features more advanced and usable by a software developer, such as allowing me to take pictures remotely from the command line, or allowing access to the camera storage as a network share?
I can't answer that, but given that the Canon app can, it's possible. But exposure of the Wi-Fi functionality through the C-language EOS Digital SDK (EDSDK) may not exist, judging by this stackoverflow question.
Also, with some fiddling that will definitely void the warranty, can I hack the firmware to run Doom on my DSLR?
Well, the Magic Lantern guys have ported Arkanoid (see also this youtube video). So, there's always a possibility, if you go Canon. ;)