If you are interested in following discussions about specific topics in mathematics, here are some blogs:

What’s New: Terence Tao’s blog: Terence Tao posts 23 times a week. His blog posts include notes for lectures in the courses he is teaching, notes on seminars where he gave talks or attended talks (such as this), announcements of published research papers (such as this), discussion of some general open problem or mathematical principle (such as this or this), or discussion about the technicalities of doing mathematics (such as this). There are occasional personal posts, such as this and this. Tao’s blog posts have been collected in two books (one for 2007 and one for 2008) published by the American Mathematical Society. A complete list of his books can be found here.

Timothy Gowers’ weblog: Contains a mix of observation and speculation on mathematical problems (such as this one on complexity lower bounds), announcing project ideas (such as this) and commenting on public policy and matters of general interest from a mathematical perspective (such as swine flu, tennis, and the ivory tower). Gowers also used his weblog to host part of the polymath projecti in FebruaryApril 2009. Gowers’ main area is combinatorics.

Gil Kalai’s weblog: This is a weblog on combinatorics. Of late, it has been used to host a polymath discussion on the polynomial Hirsch conjecture.

The nCategory Café: A group blog on category theory and its relationship with mathematics. Some of the blog posts are in other areas of mathematics, and some blog posts are related to science policy. The contributors are John Baez, David Corfield, Alexander Hoffnung, Tom Leinster, Michael Shulman, Urs Schreiber, and Simon Willerton.

Secret Blogging Seminar: Describes itself as “A group blog by 8 recent Berkeley mathematics Ph.D.’s. Commentary on our own research, other mathematics pursuits, and whatever else we feel like writing about on any given day. Sort of like a seminar, but with (even) more rude commentary from the audience.”

Quomodocumque: a blog by Jordan Ellenberg: A mix of blog posts, a large fraction of which are mathematicsrelated.

Noncommutative geometry, a group blog that covers topics in noncommutative geometry, includes conference announcements and information about conference proceedings, and sometimes has more general content on mathematical philosophy.

Rigorous trivialities: A group blog by Charles Siegel, Matt DeLand, and Jim Stankewicz.
Also of interest may be the AMS Grad Student Blog.
For a longer list of mathematics weblogs, see the Mathematics/Statistics page on Henry Farrell’s wiki or this list on nlab.
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