During character creation stage, select the "Import ME2 Character" option to import your Mass Effect 2 character into Mass Effect 3

Note: You MUST have completed Mass Effect 2 in order to be able to import Shepard to Mass Effect 3

After you select the import option, it will say:
"Searching for characters to import"

You can NOT import Mass Effect 1 files directly into Mass Effect 3. You can only use Mass Effect 2 characters.

Also note, that based on Shepard's actions and decisions in Mass Effect 2, there is a chance that Shepard died in ME2 and did not survive. If in your save game Shepard is dead, you will NOT be able to  import Shepard into ME3.

If you do not have a character to import from a completed ME2 game, or you choose to start a new character for ME3, the game will prompt you on some major decisions from ME1 and ME2, including whether you grew up as a Spacer, Colonist or Earthborn, if you were a Sole Survivor, War Hero or Ruthless,  and whether Lieutenant Kaidan Alenko or Gunnery Chief Ashley Williams died and which one survived.

After you have imported Shepard from Mass Effect 2, you will be allowed to make changes to Shepard's appearance. It is also possible that the face-file of your ME2 Shepard is not compatible with Mass Effect 3, for various reasons.  You will see the message:

"Mass Effect 3 cannot successfully determine the custom face code used by this imported save game. Please update your character's appearance."

In this case, you will have to recreate Shepard's facial appearance.

You will also be allowed to change Shepard's class at this stage. For example, if you played a Soldier Shepard in Mass Effect 2, you will be able to change him to Adept.

On successful import, the game will display your decisions and choices from the previous Mass Effects that will affect the story in the current Mass Effect. Some examples:
Saved the Rachni Queen
Killed the Rachni Queen
Rescued Ashley Williams on Virmire
Rescued Kaidan Alenko on Virmire
Chose to save the Council
Chose to Sacrifice the Council
Saved Maelon's Data
Destroyed Maelon's Data
Kept Wrex alive on Virmire
Had Wrex die on Virmire
Chose to rewrite the Geth Heretics
Chose to destroy the Geth Heretics
Destroyed the Collector Base
Left the Collector Base to Cerberus
Survived the suicide mission with 12 squadmates
Romanced Liara before ME2
Did not romance anyone in ME2
Romanced Miranda in ME2
Romanced Jack in ME2
Romanced Tali in ME2
Did not romance anyone in ME2

Troubleshooting

PC:
First, Make sure the latest patch is installed. Right click on the game in Origin, and select "Check for Update" to manually download and install the Mass Effect 3 Patch. The patch has some fixes for the facial import bug.

If the game cannot locate your Mass Effect 2 save files, make sure they are in the correct directory. You will not be able to specify in what directories the game will search, so you will have to move your save files to the default directory if they are not there already.

The default directory is:
Documents\BioWare\Mass Effect 2\Save

The directory could look something like:
C:\Documents and Settings\USERNAME\My Documents\Bioware\Mass Effect 2\Save

If you played Mass Effect 2 on Steam and not Origin, you should still be able to import your Steam ME2 save into ME3 with no problems.

If that directory does not exist, then create the new directory. Under Documents\Bioware, near the Mass Effect 3 Folder, create a new folder and call it "Mass Effect 2". Inside this new folder, create another new folder called "Save". Inside, move your save files.

Mass Effect 2 save files look something like this:
John_10_Vanguard_110310
Jane_13_Soldier_100811

Xbox 360:
There is an issue that might prevent importing your Xbox 360 save from the second Mass Effect into Mass Effect 3. ME3 does support cloud saving for new or imported ME2 saves, but it does not support importing endgame saves directly from the cloud. This issue can also cause a problem in recognizing files that have been transferred from the cloud to a console which is different then the original console from which they were created.  The following actions will result in unsuccessful importing:
1) Attempting to move your Mass Effect 2 save from your hard disk to the cloud
2) Attempting to move your save to a different Xbox 360 system using the cloud

If you want to move your save game from one Xbox 360 console to another, do NOT use the cloud. Instead, use a transfer cable, a Xbox 360 memory card, or a formatted USB stick.

If you have already moved your ME2 save from your hard drive to the cloud, the solution if you want to import it is to move it back to the Hard Drive of the Xbox 360 console that it was created in.

If you have moved your game already to a different Xbox 360 and wish to import it, the solution is to login to the Xbox 360 profile which has your ME2 cloud save, and launch Mass Effect 2. An import process will start and the save files will be downloaded from the cloud into the console's local cache. After that is done, proceed to launch Mass Effect 3 and import your character the normal way.

If none of the solutions above work, you can try replaying the final mission of Mass Effect 2, and once beaten it will create a new save that can then be imported.