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The original logo was scanned in and was pretty poor quality. You can see that there is a lot of distortion from a low-quality scan and some hefty jpg compression, resulting in both background and logo being effectively textured rather than block colours, and the edges not exactly crisp. My recommendation when dealing with logos is to use a vector graphic program like Illustrator, as this will ultimately give you a lot more control later - the finished product here is completely scalable to any size, which is what you want from a logo.
Step 1: Open in Illustrator
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This is just a blank document, with the original image placed in it.
Step 2: Trace
Select the image and go to Object -> Live Trace -> Tracing Options. This will bring up a dialog box giving you lots of options to tweak to get the desired effect. The LiveTrace tool essentially looks at your raster (pixels) graphic and tries to match paths to it to create a vector version of it. I would recommend ticking the Preview box so that changes you make to the trace settings are shown in the document there and then.
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For this example there are only two colours, so I'm going to select "Black and White" mode, as this will allow us to deal with just two shades (we'll colour it back in later). At first you may find that it all goes either completely black or completely white; you'll need to play around with the "Threshold" value to get it to find the edges of your logo accurately. The threshold is the level at which it determines the difference between background and foreground, just so you know.
Other things to play with (I say 'play' because this isn't a precise science, it's more of a trial-and-improvement thing) include Path Fitting, Blur, possibly Corner Angle. These all control various aspects of how the paths are determined. Rather than going into lengthy detail about how each one works, just play with it until you produce what you want. It'll probably be different for each original image anyway.
When you're happy with the path it's created, click the Trace button.
Step 3: Colouring it in
Use the LivePaint tool in the toolbox to give a colour to each part of the logo. First of all I got rid of the solid background by colouring it transparent.
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Next, select the colour for the logo itself, and click each bit of the logo in turn until it's all coloured in. When you hover over it the paths will be highlighted so you can see where everything is even if it's not coloured in yet.
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Step 4: There is no step 4
That's it, actually. You can now use your vector logo however you want, at any size you fancy.
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Optional Step 5: Refining the path by hand
If you find that the LiveTrace hasn't quite got it perfect enough for your liking, you can always convert it into a normal object by selecting Object -> Expand, and clicking OK in the dialog box. You won't be able to go back and do LivePainting any more, but you will be able to manipulate the path yourself with the little handles, as you would any other vector graphic.
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As an accompaniment to this tutorial, I'll also be doing one for Photoshop to achieve much the same sort of thing. Watch this space...
