TL;DR
1. Give an accurate, detailed description of your issue
2. Provide the crash log, (if applicable)
3. Provide system specs.
4. Provide Java version used/installed.

Give an accurate, detailed description of your issue
Remember, the more information you give us to begin with, the easier it will be for us to find solution to your problem. We are not psychic, so you have to tell us what is actually wrong with Minecraft, or how are you having problems with running it.

Did it give any error prompts? Did it just close on its own? Did it happen before? Has Minecraft ever worked before on the machine? What unusual behavior is it exhibiting? Those sort of questions.
All of these are very helpful in finding what is actually the problem, and that we can pinpoint the cause and hopefully get you a working solution.
It would also help if the title of your thread gives a little insight of your issue, and the Minecraft version you are running, a good example of it is as such.

[1.5.2][Windows] Bad video card drivers
[1.5.1][Mac] Cannot get JAWT interface[/b]

Provide the crash log
The crash log is the most essential part of trying to figure out what went wrong with Minecraft, or your computer in general, there are 2 types of crash logs.
Minecraft crash log (Looks like this)
Java crash log (hs_err_pid.txt file that shows up on your desktop)

There are times where there will be no crash log, and that the Minecraft game screen will only appear black and/or white. For that you would have to grab the crash log manually, more info on that on this wiki page.
However you have obtained your crash log, it would be wise to either spoiler it, or paste it on pastebin then share the URL on your support topic so it would not stretch the page.


Provide your system specifications
A. Windows
The DxDiag log provides us with your full and accurate system specs so you do not have to guess what is onboard your computer. Like crash logs, this is very long, it would be wise to either spoiler it, or paste it on Pastebin then share the URL on your support topic so it would not stretch the page.

B. Mac
If you are on a Mac, please specify
What year is it from?
What is it? (A Macbook, Macbook Air, iMac, etc)
What version of OS X is it running?

C. GNU/Linux
If you are on a Linux system, please specify the general info below. You can also obtain this information through the Terminal.
Your CPU
Your GPU
RAM installed
Your distro

Specify your Java version
On most operating systems, you can open up Command Prompt, or Terminal then type in this command,
java -version
The output would show the version and architecture(32-bit or 64-bit) of the Java you have installed. For example,
In this case, it means that I have Java 1.8.0 (Java 8) installed, and it is the 64-bit version.

First and foremost, be sure to follow Microsoft-Mojang's Terms of Service as well as their End User Agreement License.

What the Launcher should look like,


This is the new minecraft launcher, these steps will not work with the legacy launcher.

Step One: Adding New
In the tab list at the top of the launcher, there's a tab called Installations. Click that and select + New...


Step Two: Name and Choose Version
Here you can name your new installation and select the version you want to downgrade to in the dropdown. Once down, click Create in the bottom left-hand corner


Now that you have done that you can return to the Play tab, select your new installation in the dropdown and you're all set.

Pixel Format Not Accelerated Crash

There have been lots of posts recently regarding the Minecraft crash
"Pixel Format not Accelerated" This crash is mainly to do with outdated graphics drivers, the crash usually shows up on older computers that have outdated hardware/software.

Mojang has instructions for a fix for this issue, please try it before posting on the forums

http://hopper.minecraft.net/help/pixel-format-not-accelerated/

if you still cannot fix the issue then post your crash and DXdiag on the forums for help