I mean that many of the clients may be using full hd screens so is it necessary for me too to develop and design websites on a full hd monitor for best Web designs? Does it really matter? my laptop has 1366x768 display on which I design so does it affect different resolutions?
No, having a fullHD screen (or any other resolution) has nothing to do with the design process, apart from being able to fit more programs and windows on a fullHD screen, and the fullHD screen might also show you what the page looks like on that kind of monitor. However, properly designed a website should function regardless of what resolution the user has, and you NEVER, EVER design for a fullHD resolution.