What is a standard driving record?
See an explanation of the 'Standard' driving record summaries issued by the New York DMV, and a PDF Driving Record Summary Description DS-242.1, which also shows a sample summary.
And if I request the summary in an office or in the mail, is there a difference?
There is no difference between an abstract that you request online, in an office, or by mail. They are certified DMV records. All include a certification statement and the signature of the Commissioner of Motor Vehicles, as well as the same safety watermark.