In the beginning there were Mail trains, Express trains and Passenger trains besides goods trains. So you also had Mixed trains, slow trains which had passenger coaches as well as goods wagons. Some of these may still exist on minor branch lines.
After Independence new varieties came along, such as the Janatha Expresses (only 3rd class) and Deluxe Expresses (with a small number of AC chair and AC I coaches).
In 1969 the first Rajdhani Express ran. A few years later we had the Jayanthi Janatha Express. These ran on a number of routes, though the only one which lasted long enough with this name was the first one between Nizamuddin and Mangalore, with a few coaches to Cochin. It still survives with a completely different route and name (the Mangala Express via the Konkan route to Ernakulam via Mangaluru Jn). Other short-lived phenomena included the DC (Diesel Car) Express which ran on metre gauge between Kanpur and Lucknow, and a few other routes on the NER.
In the late 70s came the “classless” trains such as the Gitanjali Express, basically all 2nd class sleeper with cushioned berths. In the mean time 3rd class had vanished and had become 2nd class just by using a bit of red paint to change III to II.
Still later, various Railway Ministers kept introducing various new ideas for express trains. No new Mail trains were introduced since the 70s. The last one appears to have been the Tinsukia Mail in c.1974, which became the Brahmaputra Mail. Some Mails were transformed into Expresses, such as the Bombay/Poona Mail to the Mahalaxmi Express (if I remember right). And the one-time 1/2 Delhi/Ahmedabad Mail first became the Haridwar Mail and then the Yoga Express (Don’t ask why. Swami Ramdev might know).
The Shatabdis, Jan Shatabdis, Durontos, Garib Raths, Double Decker, Jan Sadharan and Sampark Kranti Expresses should be familiar enough now. I must have missed out a few species. Right now the new Railway Minister has unleashed several new beasts, for which you can refer to this news item:
and this older reference: https://en.wikipedia.org/wiki/Express_trains_in_India
The new timetable from October 1, 2016 will include these:
Deendayalu appears to be a type of coach rather than a train.
So now you have it. India does have the most variety of trains of any railway system in the world.
Just waiting for Uday Expresses to run to these places:
As you would guess, the “original” Udaipur is the one on the right while the one on the left is in Tripura (where the local language is Bengali).
The original Udaipur station existed until 1964 when the new route from Udaipur to Himatnagar was built. It now goes by this name: