Lighting – the type of this light bulb

It looks a bit like this - there are a number of options for this type of bulb, the length and wattage can vary.

Probably your best bet would be to take the burnt bulb to a big box home improvement store. If you have a local lighting specialty shop, they can probably also help.

The holes in the white part look like they might fit a G4-base halogen bulb:

It also looks like there is a label that's curling away from the side of the fixture in the lower photo; if so, it should have information on the maximum wattage that you can use.