Q: Mac cannot start up Yosemite

Please download and install EtreCheck from http://www.etresoft.com/etrecheck

Run it and post the report here

1. Reboot the Mac and hold down Command+R to boot into Recovery
2. At the Mac OS X Utilities screen, select “Disk Utility”
3. Select the boot volume or partition from the left menu  (Usually labeled "Macintosh HD") and click on the “Repair” tab
4. Click on “Repair Disk” to repair the boot volume
5. Repeat the repair 3 times until you receive the message "Volume appears to be OK" 2 times in a row.

You could try & see if the computer may eventually start into

Safe Mode or Safe Boot, where you'd hold the Shift key down

for a long time just after startup, and until there is evidence of

the Finder window appearing. Sometimes this can take up to

15 to 10 minutes, depending on several factors.

•OS X: What is Safe Boot, Safe Mode? - Apple Support

Once into the system, other tests and perhaps Console error

logs from the System could be viewed. You could also try &

see if Disk Utility could repair permissions or repair disk, once

you have the computer running. You may be able to start up

into OS X Utilities on the Recovery mode, to check/repair HDD.

If you get as far as a Gray Screen, and no further, this may help:

•Mac OS X: Gray screen appears during startup - Apple Support

•OS X: About OS X Recovery - Apple Support

More complex & advanced methods of attempting to diagnose & repair

issues on startup can be rather involved and some of the methods do

require a high or greater degree of understanding the Terminal or the

command-line access to the underpinnings of OS X. This goes behind

the scenes where the user interface is not simple or user-friendly...

For those who have read & carefully tried the command line access

and not cause more damage, once learned it can be helpful & direct.

•Mac OS X: How to start up in single-user or verbose mode - Apple Support

Use of command line and Terminal access is not recommended in

general by those who have a problem and only one computer to

learn how to use it; you may have a worse situation due to lack

of experience. So other methods less invasive (disk utility) are the

recommended path; or if problems point to hardware cause, then

the computer would best be taken to an Apple Specialist or genius.

•Resolve startup issues and perform disk maintenance with Disk Utility and fsck - Apple Support

So if you can access the computer to get the basic report mentioned

in the above post (etrecheck) that may show someone some clues as

to what may be wrong. However a command line report such as that

which uses terminal commands, is good if read through by someone

who understands what the cryptic details mean.

Hopefully you can post more information because there are several

things that may contribute to your computer's symptoms...

To troubleshoot involves attempting several things to see what works
when working in the blind on resolving problems by rote. However if

you haven't experience in the Terminal or command-line, tread careful.

Good luck & happy computing!

edited

The startup drive is failing, or there is some other internal hardware fault.

Back up all data on the drive immediately if you don't already have a current backup. There are ways to back up a computer that isn't fully functional—ask if you need guidance.

Make a "Genius" appointment at an Apple Store, or go to another authorized service provider.

If privacy is a concern, erase the data partition(s) with the option to write zeros* (do this only if you have at least two complete, independent backups, and you know how to restore to an empty drive from any of them.) Don’t erase the recovery partition, if present.

Keeping your confidential data secure during hardware repair

Apple also recommends that you deauthorize a device in the iTunes Store before having it serviced.

*An SSD doesn't need to be zeroed.