The main official portal of Indian Railways (http://www.indianrail.gov.in/) has a special section where you can check trains between important stations. You can also check the availability of tickets for trains listed for an enquiry.
For those who have an IRCTC account, they can check the trains between any two stations on the website using the station name or station codes and book tickets instantly. Enquiries for train availability will show all trains listed for a particular date, including special trains.
A number of other websites offer facilities regarding to checking train tickets and finding the train status, but none of these sites will help in booking because of sole authority of IRCTC. You can check in with a website of your preference and check other details, except ticket bookings.