I have noticed people here frequently talking about "Illustrator(s)" and illustration. I have no idea what 'illustrator' means in this sense!
What is an illustrator? How is an illustrator different to a designer?
Illustrators are people who do illustration, like how designers are people who do design.
If your question is essentially "What is 'illustration', and how is it different to graphic design?", that's a good question. It's a difficult one to define. Most people just have a gut feeling for where the line is between graphic design and illustration. My understanding has always been:
Illustrators and designers use similar tools and have similar skills and training. Most illustrators are competant designers and most designers are competant illustrators. There's a lot of overlap between the two disciplines. It's common to see courses with titles like "Graphic Design And Illustration".
In terms of skills involved, illustration is generally more expressive, art-y, about images that create a subjective mood, and graphic design is more deliberate, science-y, about images that achieve an objective goal.
Here's some examples taken from Threadless.com, a clothing site with designs made by illustrators and also by designers:
This t-shirt is an example of illustration. It has a really strong expressive style and creates a mood bringing to mind bikes, chaotic things and energy without you needing to look at it directly. It doesn't need your attention to work and, when you look at it directly, it doesn't try to direct your attention around itself - your eye wanders around freely.
This t-shirt is an example of graphic design. It attracts attention to itself with a strong, clear focal image, and uses a visual hierarchy to pass attention down to the smaller image below, which carries a deliberate meaning about bikes not needing fuel. It works with your attention in a controlled, deliberate way.
This t-shirt is somewhere between the two, both graphic design and illustration. It creates a whimsical mood even if you don't look directly at it, attracts and directs attention in a deliberate set way (but it doesn't demand attention), and it communicates a deliberate message if you do pay direct attention to it.
A lot of (most?) design work is somewhere between the two like this. Illustrations often carry a deliberate meaning, usually a reframing of the meaning of the thing that it illustrates, for example illustrations in children's books which visually show the action that is being described in the text. A good children's book illustration adds depth to and enhances the very simple text without competing with it. Graphics are often designed to give an abstract impression of mood and personality (usually in a more carefully controlled, analysed and deliberate way), for example a good corporate logo has to meet a lot of very specific requirements about how it is to be used and function, as well as giving the impression of an abstract 'personality' of the company.
It's a very fuzzy line between the two disciplines.
Adobe Illustrator = a particular vector based illustration program made by Adobe.
Illustrator = a person who illustrates. Typically in the world of Graphic Design it would be a person who takes concepts from art directors or editors and creates a visual via drawing, painting or what have you.
I'm a total noob here so not sure. But I think Illustrator is referring to vector based graphics programs that create images based on mathematical formulas and therefore scale very well (e.g. Adobe Illustrator), (because everything just has to be enlarged or decreased by a numerical factor).
The alternative is raster-based programs (e.g. Adobe Photoshop) which modify individual pixels and are therefore very good at working with things like photographs which have lots of slightly different colors.
In almost every way, I agree with @user568458 so I have very little to add, except this excerpt from the book Illustration, a theoretical and conceptual perspective, by Alan Male.
I will also recommend The art of looking sideways by Alan Fletcher