I am totally confused between two terms i.e.Data connectivity and API's. If Data connectivity allows the servers and clients to connect and transfer data between them then where API's comes into play?

Data connectivity is the ability to connect clients and servers, and that ability is possible through the use of API's. Here is an example - imagine that a server is an ATM machine and a client is a person who wants to withdraw money from it. The API of the ATM are the buttons, the screen, the slot where you put your card in, and the slot from which the money comes out. The data in this example is the card, the information you enter(such as the amount you wish to withdraw, your pin) and the money that comes out, and the data connectivity, the ability to transfer that data between you(the client) and the server(the ATM), is made possible through the use of the API of that ATM(its buttons, screen slots, ...).