1. A method of operating a client processing system that includes one or more client processors, a character input mechanism, and a display, the method comprising: (A) accepting from a user a request for a Web page of a website; (B) sending the request to a server processing system coupled to the client processing system, wherein at least one of: the server processing system comprises a source of websites, and the server processing system is coupled to a source of websites, wherein responsive to receiving the request, the server processing system: provides the requested Web page of the website or another Web page of the website, ascertains whether the requested Web page or website thereof is included in a pre-determined answer data structure stored in or for the server processing system, and upon ascertaining that the requested Web page or website thereof is included in the pre-determined answer data structure, modify the provided Web page by adding a search user interface element (search UI) for display with or in the provided Web page; (C) accepting the server processing system-provided Web page of the website including any search UI added by the server processing system for display with or in the provided Web page; (D) displaying the accepted Web page on the display, including any accepted search UI added for display with or in the provided Web page; (E) thereafter: (E1) accepting, one by one, one or more characters of a search query from a user via the character input mechanism; (E2) sending the accepted characters on-the-fly portion-by-portion to the server processing system as the user enters the characters, such that at any time, the sent characters from the user form an initial part of one or more queries and would eventually form all of a query for carrying out a search via the website; and (E3) receiving from the server processing system, in response to the sending of each portion, a list of answer entities that are possible answers to the one or more queries whose initial part matches the so-far-sent initial part, wherein the list of answer entities is a result of the server processing system's looking up the pre-determined data structure of pre-determined answer entitles to the one or more queries whose initial part matches with the so-far-sent initial part, and wherein for each pre-determined query of a plurality of pre-determined queries on a plurality of Web pages of websites, the pre-determined data structure of answer entities comprises: one or more pre-determined answer entities for the pre-determined query, and for an answer entity residing in a Web page that is in a hierarchy of Web pages, the context of each answer entity for the query, the context of the answer entity comprising a parent answer entity that includes the title of the parent Web page in the hierarchy and a link on a parent Web page; and (E4) displaying the list of answer entities together with the context for each answer entity that is stored with its respective context in the data structure of answer entities, wherein each answer entity in the list includes a selection mechanism, such that the user may select a particular answer entity from the list, wherein the list of answer entities may change as a result of the user entering each additional portion until the complete query is entered, and wherein an answer entity for a query is a Web page or portion thereof that is an answer to the query.