Ok, I see pretty often people asking for vector-tutorials so I decided to collect some of them. First of all you need a application designed to handle vector-graphics like for example Adobe Illustrator, Macromedia Freehand MX or Flash MX.
There are also free open-source applications like Sodipodi and Inkscape.

If you dont have the possibility to use any of those you can use Photoshop, but you wont have the same advanced tools as the applications designed to handle pure vector graphics. Secondly you need to start practice, so read the tutorials below and have fun!

Vector Tutorials:

*CrisVector made this:

Dragon Girl - Tutorial (skin, eyes, hair, gradient meshing, use of pathfinder and other tools)

~novenarik made this:

Flesh Tone Tutorialish (good skinshading technique)

*tikideputy made this:

Walkthrough Tutorial (basic illustrator skills, portrait tips, common problems)

~yajido made this:

CS Vector Tutorial 3 (skin, lips, eyes, hair)

=rockst3ady made this:

ENGLISH Vector-Freehand Tut. (drawing shapes, coloring)

~jhonsyk made this:

Vector Detailed Hair (drawing detailed hair)

~christinee made this:

Gradient Mesh Tutorial (gradient meshing)

`lemontea made this:

Lemon's Vector Guide (general PS, CS, CS2)

=ChewedKandi made these:

3 tutorials for Paintshop Pro 7 vexellers (how to make vectorstyled raster-art)

and this:

tutorial for Illustrator (hand, neck, top, jeans, face shading, facial hair, lips, hair, eyes)

~lucky13 made these:

4 tutorials for Photoshop and Illustrator (gradients, swatches, symbols, pen tool)

*PamzyLove made these:

tutorials for making brushes and other stuff in Illustrator (brushes, various)

=minimonster777 made this:

tutorial for absolute beginners (the basics)

~sentinel2001 made this:

Vector Eye Tutorial

^zilla774 made this:

Sugar Bag Tutorial, Pepsi Can Tutorial and Celtic Knot Tutorial

~Excel-Mp made this:

VectorGirl in Illustrator-Tips

~kapopopoy made this:

Tutorial-102 Adobe Illustrator

*Derian made this:

Modeling A Basic Wine Glass In Adobe Illustrator CS/CS2

~LouBlue made this:

Gradient Mesh Tutorial

**********Update10: Various tutorials added
*********Update9: More tutorials by *PamzyLove added
********Update8: Added another vector tutorial by ~lucky13 and a tutorial for absolute beginners by =minimonster777
*******Update7: Added =ChewedKandi's vector tutorial.
******Update6: Added *PamzyLove's brush tutorials.
*****Update5: Readded ~lucky13's tutorials (+1 new).
****Update4: Added =ChewedKandi's vexel tutorials (for PSP).
***Update3: Added `lemontea's tutorial.
**Update2: Removed ~lucky13's tutorials since he decided to leave DA in protest of the treatment of °jark. Added *tikideputy's tutorial.
*Update1: Added various tutorials.