I was updating my macbook air to Yosemite. When I almost finished the update, my battery died. I've charged my laptop, but it won't turn on again. The screen just stays black. What should I do?
During Installation Yosemite also installs a Recovery partition on your disk.
You might be lucky and it has finished that job before the lights went out.
You could try to call it up by restarting while holding cmd + r
If that did not work, you will need a external Yosemite.
Using a USB stick with bootable Yosemite you can reinstall it on your Hard Disk.
How to get it?
From another Mac, from a friend, from Apple Store...