How to tell if this light's bulb is blown? And what kind of light is it?

There's a circuit board inside the white plastic bit, which is called an LED driver.

Here's an example http://www.dx.com/p/3-6v-16v-925ma-constant-current-led-driver-board-for-cree-and-ssc-leds-4pcs-506226#.W26FJ3b-hhE which looks like this:

The red and black leads go to the LED and the two round metal tracks on the other side are for interfacing to supply battery. The specs relate to voltage out, and supply voltage available as well as maximum currents.

Given your light is not working now, you're not going to make it worse with some exploratory surgery.

Now some observations:

• There is no thermal conductor around that unit. Its all plastic. Normally a high-power LED makes a lot of heat that must be removed by passive heatsinks.

• That LED is... tiny. Normally bike lights come with MUCH higher power LEDs that look something like this: Note the complete aluminium back to allow heat transfer to a heatsink?

So if your light isn't obviously damaged on inspection (loose wire or leaked battery or corrosion) then its probably best to junk the light and buy something newer. Its done its service, time to retire it.