1. An electronic settlement system for effecting an authentication through a communication network, comprising: a first terminal for purchasing an item by a user thereof, which is at least one of a cellular phone and a PDA, the first terminal including an input unit for inputting authentication information of the user and connecting to the communication network; a second terminal for charging the user of the first terminal a purchase amount, the second terminal being connected to the communication network; a database for storing authentication information of the user and a plurality of authentication methods; and a mediating server which performs the settlement of the authentication by mediating a communication between the first terminal and the second terminal one-to-one when receiving an ID information from one of the first terminal and the second terminal so as to determine that the first terminal and the second terminal are participating in a same purchase, the mediating server setting at least one of the authentication methods selected by either one of the users of the first terminal and the second terminal, the selected authentication method being processed between the first terminal and the second terminal that have been determined to be participating in the same purchase, wherein one of the mediating server and the second terminal authenticates the user by using the authentication information stored in the database, wherein when receiving a request signal from one of the first terminal and the second terminal, the mediating server sets up and transmits the ID information to the one of the first terminal and the second terminal which sent the request signal to the mediating server, and when receiving the same ID information from the other one of the second terminal and the first terminal, the mediating server mediates the communication with the first terminal and the second terminal, the settlement of the purchase is executed by an instruction from said first terminal during mediating the communication with the first terminal and the second terminal, wherein the request signal includes a unique ID of at least one of the first terminal and the second terminal, and wherein said database includes, a first terminal database for storing the authentication information of the user and the authentication methods demanded by the user of said first terminal, and a second terminal database for storing an authentication method demanded by the user of the second terminal, wherein the mediating server sets an agreeable authentication method in accordance with the authentication method stored in the first terminal database and the authentication method stored in the second terminal database.