Vector art is my favorite weapon in attacking graphic design. Vector art can always be edited unlike raster art which is pixel-based. Photoshop is a raster based editing solution. Adobe Illustrator is an all in one solution for your vector art. The cool thing is they can be used in conjunction with each other back and forth.
For instance you can make your original file in Photoshop bring it into Illustrator and then trace it with vector lines to give you a scalable graphic. Below is it tutorial that will get you familiar with Adobe Illustrators tool panel.
Adobe illustrator has a lot of applications in the graphic industry. It is best for graphic design, t shirt on is great for motion graphics or anytime you need to make your design for print. I even use it for animation to create each each moveable asset as well as backgrounds or any other vector element.
When making a logo for a client I like to start with pencil and paper then scan image in and imported directly into Adobe Illustrator. From there you can use Live Trace tool or manually create the outlines for the image. From there you can easily copy and paste your graphics and adjust on the fly to get proofs to your client which they can choose from. All of these images on out edible and printable.