Adobe Illustrator has some powerful tools to shape and modify display type (type that is used in headlines etc.) I don’t recommend it for laying out body copy however.
When discussing typography, it is necessary to understand the terminology. Here is a wonderful site that has an extensive list of typographic terms. Once you are familiar with some of the terms you can begin to compare typefaces to see how they differ.
Here is a fun way to get familiar with the art of letter spacing or kerning.
But typography is more than just dissecting a typeface into its component parts. When used well, type can add meaning to the text. It can also be expressive, although some old school typographers reject the use of type in this way.
The following clips are from the movie ‘Helvetica’ by director Gary Hurstwit. The movie is ostensibly about the typeface that defined modern graphic design, but it also tackles some of the issues surrounding graphic design in general.
Stefan Sagmiester has turned typography on it’s ear with this series.