You are correct in your understanding of the DOM based model. The XML file will be loaded as a whole and all its contents will be built as an in-memory representation of the tree the document represents. This can be time- and memory-consuming, depending on how large the input file is. The benefit of this approach is that you can easily query any part of the document, and freely manipulate all the nodes in the tree.
The DOM approach is typically used for small XML structures (where small depends on how much horsepower and memory your platform has) that may need to be modified and queried in different ways once they have been loaded.
SAX on the other hand is designed to handle XML input of virtually any size. Instead of the XML framework doing the hard work for you in figuring out the structure of the document and preparing potentially lots of objects for all the nodes, attributes etc., SAX completely leaves that to you.
What it basically does is read the input from the top and invoke callback methods you provide when certain "events" occur. An event might be hitting an opening tag, an attribute in the tag, finding text inside an element or coming across an end-tag.
SAX stubbornly reads the input and tells you what it sees in this fashion. It is up to you to maintain all state-information you require. Usually this means you will build up some sort of state-machine.
While this approach to XML processing is a lot more tedious, it can be very powerful, too. Imagine you want to just extract the titles of news articles from a blog feed. If you read this XML using DOM it would load all the article contents, all the images etc. that are contained in the XML into memory, even though you are not even interested in it.
With SAX you can just check if the element name is (e. g.) "title" whenever your "startTag" event method is called. If so, you know that you needs to add whatever the next "elementText" event offers you. When you receive the "endTag" event call, you check again if this is the closing element of the "title". After that, you just ignore all further elements, until either the input ends, or another "startTag" with a name of "title" comes along. And so on...
You could read through megabytes and megabytes of XML this way, just extracting the tiny amount of data you need.
The negative side of this approach is of course, that you need to do a lot more book-keeping yourself, depending on what data you need to extract and how complicated the XML structure is. Furthermore, you naturally cannot modify the structure of the XML tree, because you never have it in hand as a whole.
So in general, SAX is suitable for combing through potentially large amounts of data you receive with a specific "query" in mind, but need not modify, while DOM is more aimed at giving you full flexibility in changing structure and contents, at the expense of higher resource demand.