Your website has a Layout Mode has one purpose, to help you rearrange the content on your page. If you go into Layout Mode, you'll notice your website will appear much different than the other modes. All you'll see is the headers for any modules on your page, and it will show you a bunch of titled boxes as to where modules can be placed.
Please pay special attention to the names of the "Panes" in this image. You can place or move modules (textboxes or any functionality on your site) to any "Pane" shown below. This allows the website to be VERY flexible when it comes to customizing layouts:
The ContentPane is the default pane and will be used the most often. If you want to do a 2-Column layout on your page, you can simply put one HTML Textbox in the Content Pane and one in "SidePane2". If you'd like a 3-Column Layout, you can put HTML Textboxes in "SidePane1", "ContentPane", and "SidePane2" to achieve this.
To move an item from Pane to Pane, simply mouseover the "Manage" button of the module you'd like to move, and select which pane you'd like to move it to in the bottom portion of the popup menu: