What is Core Banking & which Banks in India offer Core Banking Service ?

Core Banking Solutions, also known in short as CBS is a system in which all branches of a bank are interconnected via computers. This will help the customers to access his/her accounts from any branch of the bank.
The CBS allows the customers of the bank to perform basic transactions easily over the networked branches of the bank. Services like withdrawals, payments, loans and transfers through a range of avenues like ATMs, internet banking and mobile banking.
CORE essentially stands for Centralised Online Real-time Exchange. The details of the accounts are stored on the centralised data centres of the bank. Thus all the transactions are reflected instantly on the bank's servers no matter from where the transaction has been made.
Some of the banks do the networking on their own, while some outsource the responsibility. There are several agencies which take up the task with the help of advanced software.
Almost all the major banks in India have Core Banking enabled. You may need to check with your bank for further details.
As in your instance, Corporation Bank does have CBS enabled system. A best way to identify whether your bank is CBS enabled is to check your account number. Unlike the conventional shorter account numbers, CBS enabled branches have a longer account numbers that run for 15 or 18 digits.

Core Banking not only allows you to perform transactions at any branch, or other medium such as internet banking, but it also helps your bank to to be interconnected with the other banks and their branches thereof both within the country or elsewhere in the world. Transactions within India are made through the channels like NEFT and RTGS, while international transactions are made possible by means of SWIFT codes.

Advantages of CBS
1. It lets the customer access banking services 24x7.

2. It is the most efficient way of performing banking transactions.

3. Faster than the conventional methods of banking.

4. From the bankers' point of view, it improves the efficiency of the services.

5. Involves greater customer satisfaction.

6. It reduces the operational and maintenance costs of the bank as you need not have more branches.

In layman terms Core banking can be explained in following way.

In first case, bank under same name are connected using centralized software system. They make use of unique codes to identity each other. Account opened at any branch of the same bank can be accessed from any other branch. This is possible due to unique identifier attached with the account names. The centralized system here is known as core banking solution.

In second case of core banking solution, the banks are given feature of NEFT and RTGS. These two are the system which allows members to transfer money from within and outside different branches of banks. NEFT allocates IFSC code for each branch to identify the home branch of the member. And during the money transfer you are required to enter IFSC code along with your customer code. And the money transfer becomes instant due to this system. In practice it takes upto 1 hour for money to reach the members account. Same is the case with RTGS, however in case of RTGS the money to be send has some strict limit and money also reaches a bit slower than NEFT.

In third case, banks are connected using SWIFT and IBAN system. Only the core branch or the first branch of the bank is allocated with SWIFT and IBAN number. Once this number is known you can transfer the money to the member account from any part of the work where SWIFT and IBAN is supported.

In short the core banking solution offers connection within branches of same bank, branches of other banks and finally banks outside the nation.

Do note that there are banks who offer core banking solution only within their own branches. And they don't have support for money transfer to other branches or offshore. Such banks are under urban category or patsanstaha and chit banks. So for them core banking solutions are limited due to government policies.