You could read the paper: Gapped BLAST and PSI-BLAST: a new generation of protein database search programs

For the papers describing the various versions of BLAST produced by NCBI, including details of the algorithms, see "Does Anyone Know How To Cite Ncbi Wwwblast In A Manuscript?". For an overview of the method used in BLAST see the Wikipedia page: BLAST

Please note that "blast" is commonly used to refer to any sequence similarity search method, and so you will need to check for details of the specific method being used in the service documentation, rather than assuming that NCBI BLAST is being used. Some examples to illustrate the diversity of options are:

• ENA Sequence Search: uses Exonerate
• Ensembl Genomes uses a range of sequence search methods:
  • "BLAST": uses WU-BLAST, as provided by the WU-BLAST (REST) web service
  • "Sequence Search": uses Exonerate, as provided by the ENA Sequence Search web service
• PDBe sequence searches: use the Smith & Waterman as implemented in the 'ssearch' program from the FASTA suite
• UniProt.org 'Blast': uses NCBI BLAST+, as provided by the NCBI BLAST (REST) web service

For a selection of some of the more commonly used sequence similarity search methods, and pointers to the associated papers, you might find looking at the selection of Sequence Similarity Search tools provided by EMBL-EBI a useful starting point.