The BLAST Home Page
Go to http://www.ncbi.nlm.nih.gov/blast/Blast.cgi
or you can get there from the NCBI home page by clicking on "BLAST" in the popular links menu.
Use MyNCBI to Save BLAST Searches
You can use MyNCBI to save your BLAST results. Click on the MyNCBI link in the upper right corner of any NCBI page to log into MyNCBI where you can manage your searches and collections.
Choosing the Right BLAST Tool
Chose your BLAST platform from the BLAST main page: nucleotide, protein or translated BLAST, or choose a specialized BLAST
Specialized BLASTs have been created to help you find vector contamination, create primers, search conserved domains, and many more choices. There's even a downloadable package for Magic-BLAST, NCBI's RNA-seq alignment tool.
Most BLAST tools can be run on the NCBI web interface (except Magic-BLAST), but NCBI has API versions, the option to run BLAST via Amazon Web Services, and downloadable code to run locally.