Along with InDesign and Photoshop, Illustrator makes up the "big three." If you need to create crisp vector artwork, either for print or screen, Illustrator is the only serious choice. Whether it's a logo, a map, a diagram, a unique type treatment, or a hand-crafted illustration, Illustrator's powerful suite of tools puts a wealth of possibilities at your creative fingertips.
When teaching Illustrator, I like to deconstruct and recreate iconic works of graphic design (most of them created long before the design software age) as a way of getting inside the creative process. While good design isn't just about your tools, having powerful tools certainly makes the execution of that design a whole lot easier.