Since all Illustrator objects are vectors it's much easier to create dynamic effects from basic shapes. Gradients can be applied onto a single shape or they can be created by blending multiple shapes together.
I'll demonstrate the latter technique using a little trick known as the Blend Tool. It's actually very simple once you learn how to tweak settings and get the gradients working as you intend.
1. Drawing Shapes
To get started open Illustrator and create a new blank document of any size. Now draw a shape like a square or circle, or even something more detailed like a polygon. Change the fill color to something other than white so you can see it.
You should also remove the stroke from the shape so it just has a solid color. Now draw another shape within this current shape – it can either be larger or smaller. If larger then you'll need to find the other shape and right-click Arrange > Bring to Front.
Now we can move the smaller shape towards the center of the larger shape and blend the colors together.
Select the Blend tool from the tools panel by clicking the icon or hitting the keyboard shortcut W.
2. Blurring the Shapes
If you want to create a blur effect manually just click within the 2 shapes to create a blurred border. You can add points to the blur effect for manipulation or manual editing.
The alternative would be in the menu bar under Object > Blend > Make.
Both techniques perform the exact same thing while keeping the two shapes separate.
If you want to edit the blur effect you'll need to mess with blur options from the tools palette. To do so merely double-click on the Blur tool which opens the Blend Options dialog box.
3. Editing Options
It should be noted that blend options are somewhat miniscule in nature. You can add extra points into the gradient or manipulate how the blending occurs. But generally speaking you'll want to practice this effect on your own in real projects to see how they look.
As you toy with different values you'll learn how to best utilize this tool. It's very different from the gradient tool because it works based on the internal shape's path.
For example, if we add another anchor point to the inner path we can manipulate the gradient's shape. This allows you to develop more detailed objects and multi-sided polygons with ease.
Keep this in mind for future vector projects when building icons, characters, or even logos. You might even try lowering the opacity on the outer shape to get a clear transparent gradient effect with path manipulation.
Illustrator is like every Adobe program in that it requires practice and patience. The more you try to build the more you'll learn and the quicker you'll see improvements in your skillset.