How can I obtain this vector effect in Illustrator? Is there a name for this kind of effect? Please, help!
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I doubt it was made in illustrator vector but it could've been. If so it was a very expensive custom job. More likely its a photo brought into photoshop then adjust levels and maybe use the posterize filter along with some hue/saturation techniques for the coloring.
To do this in illustrator you just need to draw all the different shades as vectors which is a lot easier said then done. There are plenty of tutorials out there if you search for "Illustrator Portrait Tutorial." The one I used when learning doesn't exist anymore. Again though to get to the level of the picture you posted takes a lot of patience and time.
Here's a tutorial that has a similar quality to the picture you posted if you do want to go for it: Spoongraphics - How to create a beautiful vector portrait in illustrator
You'll notice the first steps in that tutorial is the things I said in Photoshop. Basically you're always going to start by simplifying your photo and getting a tonal quality you want in photoshop. If you then really need it in vector you have to bring it in to illustrator and start tracing.