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How to Check Seat Availability in Indian Railways
Create an account on http://www.irctc.co.in
After creating the account, login into the site. There will be a form on upper left hand side, Plan my travel. Fill the form, Destination City, Departure City and Date of journey, tickety type and quota type.
The site will suggest the list of all trains between the given stations on the mentioned day along with timings, pricing and seat availability.
The screen will be showing all the trains with radio boxes, you can select any radio button and the screen will display the number of seats available. You can click on the book to reserve the tickets.
For tatkal quota, the screen may show seats available but they can only be reserved two days before the departure of the train, so if the train leaves on Saturday, the tatkal tickets can only be booked on Thursday.
Other Sites to Check Seat Availability in Indian Railways
There are many online travel booking sites who show ticket availability online. They are using IRCTC real time database and API, some of such sites are

Ixigo.com
Yatra.com
If IRCTC site is not working, you can check these sites to get seat availability in Indian railways.