A fixed (or static) design basically means the parent container (the body of the website more or less) is set to a static width. That means that no matter if you're viewing the site from a phone, a tablet, or a PC, you will get the same width displayed to you.
Fluid web design means that the width of all elements (including the container) will have a percentage width. That means it will adapt to the screen resolution of its user. Its child elements typically also have percentage widths, however those are related to the width of the parent container.
Responsive page layouts are layouts that are specifically designed to work well in any given environment (PC, phone, tablet). Typically, they combine the use of fluid web design, with media queries to further help. An example is the "Breadcrumb menu", which is commonly used today. In a small screen, the menu of the website is compressed into a clickable button that displays it. This is to help display the more relevant content without the menu being a disturbing factor.