Can I watch 3D movies on my computer just using 3D glasses and my normal monitor or do I need a special monitor to see 3D?
3D used to be made using red/blue (called anaglyph), which any monitor can do, but that's old and no longer used.
Modern 3D uses switching and timing, where the image for one eye is displayed briefly and then for the other eye next. So-called shutter glasses are required because they'll switch between the eyes at the same speed as the screen. Or it uses polarization (a simpler system more akin to red/blue from yesteryear) to separate the two images.
In other words, playing a modern 3D movie requires a special screen (no software can make your normal monitor do switching or polarized images) and probably glasses. For example, the screen needs to support a refresh rate of 120 Hz, and you need a pair of glasses like NVIDIA 3D Vision. You will also probably need a graphics card that supports dual-link DVI in order to attach the screen.
Here's a site with more details: http://www.makeuseof.com/tag/laptop-monitor-tv-3d-frequently-asked-questions/
Contrary to what music2myear reports, anaglyph (blue/red and other variants) 3D is still widely available. It has certain disadvantages (such as loss of color fidelity and light) but it also has several advantages, such as no flicker and, in your case, it is usable on your ordinary monitor. One of the major video sharing sites that supports viewing of 3D content in anaglyph (and other formats) is none other than YouTube - see here for more details.
So, the correct short answer is "Yes" - you can view 3D content on your existing monitor by using Anaglyph 3D glasses - and you have a ton of readily available content on YouTube and other sites.
OTOH, if you buy a 3D DVD, you'll need to be sure it is compatible with whatever monitor you own. Most current DVDs sold are NOT designed for the anaglyph user. As m2m mentions, some options are time-sequential (shutter glasses), polarizing glasses, as well as a number of autostereoscopic (no-glasses) options, such as those produced by Sharp (parallax barrier) or other companies that use lenticular layers over an LCD screen (e.g., Philips, StereoGraphics, etc.).
If you've read this far - perhaps you'll also be interested to know that lenticular uses the same concept as those 3-D postcards you may have seen and that have been popular for 50 years or so. The disadvantage of Lenticular and Parallax Barrier is that they sacrifice resolution - you get 3D but each eye sees a half (or worse) resolution image. OTOH, you don't need glasses. Most systems that use polarizing glasses on LCD monitors also sacrifice resolution (e.g., they divide the screen real estate into left & right images & use the glasses to make only the relevant image visible to each eye). Their advantage is that polarizing glasses are usually lighter, more comfortable & cheaper than shutter glasses. Also there is no "sweet spot", like most autostereo systems, so multiple viewers can watch 3D at the same time. There is also a company called 3ality (not to be confused with the later-founded and unrelated 3ality Technica) that has developed display technology that is glasses-free and full-resolution - however it has not yet been brought to the market.
Just to jump in, but not all 3D technologies degrade image quality. Blu-Ray is using sequential 3D which sends images at full resolution at 48 frames per second alternating between the "eyes" for each frame. Most of the SBS, or OU formats do split the resolution in half unless you have FULL SBS or OU which is just a huge resolution image to hold both frames at full resolution.