How much power can be supplied over Beaglebone Black's USB?
The BBB uses a TPS2051 USB Power Switch. It limits the USB Host port power to 500mA on the high end. The power is drawn from the same system bus that powers the rest of the 5v parts, the 5v input port.
The BBB needs 1A for normal function + a low current usb device (keyboard, mouse, things that draw little current). Using a Cape or a high current USB device would require a power supply that can meet or exceed that extra current draw.
Unless you know explicitly otherwise, assume the maximum power is 100 mA.
If your external device isn't just using the USB interface for power, and is actually enumerating to the beaglebone properly, it can request up to 500 mA under standard USB.
If the beaglebone supports some of the non-standard USB-2.0 high-current charging profiles, any device that also supports the non-standard charging profiles should just work, or fall back to only 500 mA draw automatically.
Anything else is technically a violation of the USB spec's behaviour.
It should be noted that a LOT of hardware out there just treats USB connections as a dumb 5V power source, which usually works (except when the host properly monitors device current draw). Hardly anyone actually implements proper host-based current monitoring, because it would break lots of non-spec hardware, and most users would just go "why doesn't your device x work with my (actually broken) shitty MP3 player/keyboard fan/stupid USB widget.
In practice, you can generally draw 500 mA even without enumerating. Drawing any more current depends entirely on the USB host, and is implementation specific.
Fortunately, in this case, the beaglebone's schematics are available, so you can look and see what the USB current-limiting mechanism is yourself.
The USB interface is on page 4 of the schematic.
Hint: The USB power for the host-port is switched by a TPS2051B