Seems to me your trouble is not understanding the concept behind object-oriented programming. JavaScript is an object-oriented language. You should probably take a quick tour of the concepts.
Briefly, though, objects are a way to encapsulate both data (the "properties" you are referring to) and functionality (the "methods" or "functions" you can ask an object to perform).
In the case of the JavaScript DOM, there is a tree of objects, where each object contains properties that are, themselves, objects. So you can have a Form object with methods like onSubmit() and properties like "elements", which is an array of form fields. Each element in the array is another object, like perhaps a TextField object or a Checkbox.
So now you know about objects. One thing about objects is that they can inherit properties from a parent class of object. For example, in JavaScript, there is an Element class and each of the form field objects is an instance of a "subclass" of the Element class. So, since Element defines the "name" property, and TextField and Checkbox and all their friends inherit from Element, they all automatically have this "name" property available too.
The term "node" refers to a particular location in a tree or graph structure. In this case, the DOM (Document Object Model) defines the types of objects that are allowed to be nodes in the tree that represents the webpage. For each webpage you visit, the browser constructs a "DOM tree", which is a big tree of objects representing each of the elements in the webpage.
Notice that HTML is naturally in a tree-like structure: the html tag contains head and body, the head tag contains title, meta, and script tags, the meta tag contains attributes like name and content. All this is arranged by the browser into a tree of objects, and that is what you are manipulating when you use JavaScript to do DOM programming.
So to recap: objects are the fundamental representation of data and functionality in JavaScript. Elements are particular classes of object that are subclasses of the "Element" class, and which represent some kind of form field. These can be found in the form.elements array, which is a property of the form object. Finally, nodes are points in the tree of tags, text, script, and other objects that make up a webpage.
Hope it helps!