What is happening in each iteration of PSI-BLAST?
PSI-BLAST is Position-Specific Iterated BLAST.

I'm probably out of my depth here, but my understanding is that the results of the first, essentially standard, BLAST search are used to create a multiple sequence alignment. This alignment is then converted to a Position Specific Scoring Matrix (PSSM). (The linked Wikipedia entry shows this for DNA sequences, but the principle is exactly the same for protein sequences.) This matrix can then be used as the probe for the next round of BLAST, and the results of that alignment can be used judiciously to improve the matrix, and so on until the results converge.

Here is a link to some slides by Stephen Altschul which go into lots of detail.