Antyodaya Express: The best a common man can get. - Vikas Kumar

Mail trains originally were trains that actually carried bags of mail to be delivered between their termini or at intermediate stations, under special contracts with the Post Office. In many cases that was in fact the main or only reason for running these Mail trains, and some of the famous mail trains got their reputation for speed and punctuality because they were accorded a very high priority in scheduling over all the other trains on the route. It was almost unthinkable in British days for a mail train to be deliberately delayed to let another train pass, and there are anecdotal stories of mails being given priority over much-needed troop and materiel trains during the world wars. Generally the trains departed in the evening after the day's postal service closed. On many well-known routes, Mail trains were introduced first, and Express trains later.
Today...
more...

mail trains don't necessarily carry mail (although many still do), and many are really quite slow, but many retain the designation and names they had from years ago. Conversely, there are many trains that don't carry the 'mail' designation which do carry mail (Bikaner Exp., Dehradun Exp., etc.).