Grammar Girl is one of my favorite podcasts for three reasons.
1) Ms. Fogerty (Grammar Girl) talks about grammar to improve communication and writing, as well as the history of interesting expressions and the ongoing development of language. She explains the more complicated reasons for a grammar point (including the proper terms for parts of speech), but then offers a simple and straightforward rule to help you remember a grammar point, as well as plenty of clear and easy-to-follow examples.
2) I don't listen to Grammar Girl every week, but instead download six or seven podcasts at a time and listen to them on long drives. As a result, one thing I especially like about the podcast is her voice. Some podcasters speak in very conversational ways, and so they speak at a more variable rate and volume. This sometimes makes it hard to hear parts of the podcast over the noise of the road. You can turn up the volume of course, but when the speaker gets excited, it can suddenly seem like they are yelling at you. However, Grammar Girl has a very "professional sound" - she speaks with genuine interest and excitement, but also very clearly, at a slightly slower rate, and at an even volume. Thus, it is easy to listen to her voice while driving and hear everything she says very clearly.
3) Last, Grammar Girl has a great website where you can search for grammar and writing help. I teach psychology graduate students, who write many papers as well as a dissertation. I often highlight grammar and writing problems in their work, then in a comment explain the problem in a sentence or two and give a link to a Grammar Girl page and podcast to help the student better understand the issue. When I hear a podcast with a good explanation about grammar or clear writing, I can easily search her site to find it later (since I'm driving when I hear it), and add the link to my list of links for students.
Again, this is one of my favorite podcasts :)