If you are viewing a 2D picture or video, both of your eyes are viewing the same picture. However, the thing that makes 3D 3D is the fact that it gives your left eye a "left eye version" of an image and your right eye a "right eye version" of an image.
The challenge for viewing 3D is twofold. First, there needs to be some way send both a left eye version and a right eye version of each image to the display. Second, the display needs to deliver the appropriate image to the appropriate eye. If the left eye sees the right eye version of an image and the right eye sees the left eye version of an image, things go inward when they're supposed to go outward and outward when they're supposed to go inward. Alternatively, if one or both eyes see both versions of the image, you get double-imaging rather than 3D.
The Three Ways to View 3D
Broadly speaking, there are three different possible ways to view 3D. These are:
One of the earliest methods to store and send 3D video was to use the anaglyph method, in which the left and right versions of the images are superimposed onto each other in different colors, usually red and cyan. The viewer then wears a pair of red-and-cyan tinted glasses to direct the proper image to the proper eye. This method reduces color accuracy in comparison with storing and transmitting the left and the right versions of images as separate images .
Initial 3D Blu-ray Disc (BD) movies used the anaglyph method and came packaged with anaglyph glasses since this was really the only widely accepted method to use . However, on Dec. 17, 2009, Blu-ray 3D was finalized . Blu-ray 3D supports the encoding of separate left and right versions of images for 1280x720p50, 1280x720p59.94, and 1920x1080p23.976 3D [5 page 33]. Blu-ray 3D discs are also compatible with older 2D-only BD players, although a Blu-ray 3D disc placed in a 2D-only player will only play in 2D . Anaglyph 3D movies will play in 3D no matter what type of BD player or TV you have, but as already mentioned you have to put up with reduced color accuracy.
Non-anaglyph 3D through HDMI
Although finding a universally agreed way to store 3D is important, for 3D to really catch on there needed to be a universally agreed way to send it as well. In response to this need, HDMI 1.4a was approved on Mar. 4, 2010. The agreed-upon methods for sending 3D are shown below :
As you can see from above, the three techniques for sending 3D over HDMI are Side-by-Side Horizontal, Top-and-Bottom, and Frame Packing.
Side-by-Side Horizontal works by taking the left version and right version of a 1920x1080i picture, squeezing each picture to 960x1080i, and putting the left version on the left and the right version on the right within a new 1920x1080i picture [7 page 11].
Top-and-Bottom works by taking the left and right version of a 1280x720p or 1920x1080p picture, squeezing each picture to 1280x360p or 1920x540p respectively, and putting the left version on the top and the right version on the bottom within a new 1280x720p or 1920x1080p picture [7 page 11].
Frame Packing is similar to Top-and-Bottom, except that the frames are not reduced in size. For frame-packed 720p, 30 pixel lines are placed between the two frames, creating a larger frame with a 1280x1470 resolution [8 page 23]. For frame-packed 1080p, 45 pixel lines are placed between the two frames, creating a larger frame with a 1920x2205 resolution .
In active 3D, the display shows the left version of an image followed by the right version of an image. Meanwhile, when the display shows the left version of an image the glasses block the right eye, and when the display shows the right version of an image the glasses block the left eye  [7 page 12].
The alternative to active 3D is passive 3D. In passive 3D, the display shows the left version and the right version of an image at the same time. Meanwhile, the glasses direct the appropriate version to the appropriate eye. Because of this approach, passive 3D has cheaper glasses than active 3D.
So, what's the catch? In order for the display to show both the left and right versions of an image at the same time, something's gotta give. In the case of anaglyph 3D, it's color. In the case of current passive 3D TVs, it's resolution. As discussed in the Resolution, Scaling, & Progressive vs. Interlaced article, interlaced video is a way to pack two moments in time into one frame. Current passive 3D displays take a similar approach, only instead of using one frame to pack two moments in time, one frame is used to pack the left and right versions. Unfortunately, this leaves only 1920x540 resolution rather than the full 1920x1080 resolution to show the left and right versions of each image.
Still, there are couple ways for passive 3D to show 1080p in full definition - they just haven't been widely implemented. One approach is to make a 4K passive 3D display. Since 4K has 2160 rows and 2160/2=1080, the left and the right versions of the 720p or 1080p image can be shown without reducing the resolution. Another approach that was being worked on by the companies RealD and Samsung was an active/passive hybrid approach , although this was called off by Samsung  .
1. Sony Introduces the Worlds First Personal 3D Viewer. Sept. 7, 2011. ©2013 Sony Corporation of America.
2. David Katzmaier. 3. How is the new 3D TV technology different from older 3D?. 3D TV FAQ. Mar. 12, 2010 6:45 AM PST. Updated Feb. 17, 2011. CNET. ©CBS Interactive Inc.
4. Blu-ray Disc Association Announces Final 3D Specification. Dec. 17, 2009 5:00 AM EST. ©2013 Business Wire.
5. White Paper Blu-ray Disc™ Read-Only Format. July 2011. ©Blu-ray Disc Association 2011.
6. HDMI LICENSING, LLC RELEASES HDMI SPECIFICATION VERSION 1.4A. Mar. 4, 2010. ©2003-2013 HDMI Licensing, LLC.
7. AMD Radeon™ HD 6800 AMD Radeon™ HD 6900 Display Technologies. Dec. 2010. ©2010 Advanced Micro Devices, Inc.
8. Sébastien Schertenleib. PlayStation®3 Making Stereoscopic 3D Games. June 10, 2010 9:48:14 AM. Sony Computer Entertainment Europe.
9. Ron Jones. First Blog - Moving to 3D. July 2, 2011. Projector Reviews. ©2011.
10. Geoffrey Morrison. Active 3D vs. passive 3D: What's better?. June 4, 2012 2:43 PM PDT. CNET. ©CBS Interactive Inc.
11. Chris Boylan. Samsung and RealD's Active/Passive Hybrid 3D TV One Step Closer to Reality. May 17, 2011 10:00:46 AM. ©2005-2013 Big Picture Big Sound.
12. RealD Inc. Reports Financial Results for Second Quarter of Fiscal 2012. Nov. 2, 2011. ©2011 RealD Inc.
13. Greg Tarr. Report: Samsung Ends RealD 3DTV Plan. Nov. 3, 2011 8:16:44 PM. TWICE. ©2013 NewBay Media, LLC.