What is the basic idea behind the renormalization group?

The main idea behind renormalization is the scale invariance of the essential physics.

Shankar's review is a rather good introduction into the subject. In particular, he makes an interesting point about the interplay between the field theories, which suffer from infrared and ultra-violet singularities, and the solid state physics where such singularities do not exist, due to the presence of natural cutoffs. Renormalization group is then essentially studying of how physical properties change with the size of the cutoff, and finding which of the are invariant to the cutoff.