This Miami Wise logo (former AFL Team) is a very simple one and we will use it to show you how it can be vectorized in less than 2 minutes. (For practice, download GIF image of this logo here)
When vectorizing logos it is very important to recognize all the objects that logo consists of. In this case, the logo consists of two circles and one object that we will draw with pen tool. It can't be easier than this. Now, draw the largest circle using Elipse Tool (L) and align it with the circle on the template (while drawing circle hold Shift key to get perfect circle).
Select the circle and go to Object>Path>Offset Path. In Offset Path window put Offset value to -20. Click Ok. Now you have two circles, identical to the template. Lock layer 1 and create new layer. Choose Pen Tool (P) and start drawing the black object. When drawing the edges of the object near the inner circle cross the inner circle a bit. Do this on both ends. Now we need to divide the black object from the circle. Unlcock layer 1 and select black object and the inner circle.
Object > Ungroup. Delete the leftover points from the inner circle and you have vectorized Miami Wise logo.
Select the inner circle and the black object (or select all) and click on Divide on Pathfinder palette. Go to Object > Ungroup and then delete leftover points. Click on CTRL+Y or View > Preview and you can see your vectorized logo.