What is this strange, linear light bulb?
Specifically that is a 78mm T3 halogen lamp.

I have one in my bathroom as well and love it because:


The bulbs are pretty cheap.  Usually a couple dollars.  I bought 10 on ebay for $20.
They are BRIGHT.  Much brighter than LED bulbs, and some fixtures can take 200 or 250W bulbs.  For a light over a mirror thats only on a few minutes a day, I dont care about power efficiency, I just want something bright.  A 100W halogen is much brighter than a 100W incandescent or any LED bulb.
The bulbs last for ages.  I have had to replace it once in the last 3 years.
The color of light is better than most LED or florescent bulbs.


There are LED versions available, but most would be too large a diameter to fit in that fixture by the looks of it.

The downside of them is that they get quite hot.  The heat is actually essential to their long life.  Be sure to check for a sticker on the fixture stating the max wattage and not go over that with a replacement bulb.