I use illustrator when wanting to preserve the quality of the image(s) being used for the task in hand. It is very important I do my best to avoid pixilation when creating website images. After all, it is said that the first bite is with the eye and if your website doesn’t look very tasty, the odds are that you driving potential customers to other aesthetically pleasing places elsewhere!
Illustrator is also very useful when I want to create text for the web, printed material used in a workplace or for creating a desktop publication.
One thing I’ve used Illustrator for recently was to create a Profile photo (and utilize other images I keep for self-promotion purposes) that I embedded on a background of Robertson Hunting Modern Tartan.  I now attach it to emails as a signature:

Illustrator can be used to produce those massive vector graphics advertisement boards that are commonplace these days.

The general symmetry of the decoration in this picture against a white background gives it a pure, clean and simple feel.

Illustrator is very powerful when manipulating text as I have demonstrated with this Christmas card. It is also the ideal software package if you are producing schematics, logos or maps.

I saved this Christmas card picture in a GIF format as there is no use of colour and the file size of the final product is therefore much smaller than a JPG equivalent.

For this particular example, I was given the remit to use no colour and only letters of the alphabet.

This picture was created in a GIF format as there is no use of colour and the GIF format is smaller in size than a JPG equivalent with these circumstances