An analogy might be fishing on a bridge in a running river, where the river is the XML:
For SAX, the fisherman is going to stand on the bridge and has decided that she/he is only going to catch salmon, i.e. let all other species of fish go on by. So, for this fisherman, the SAX parser catches only the "salmon" elements as they go by.
For DOM, the fisherman is going to run a giant net across the river, and catch every fish in the river.
That's why it's sometimes said DOM is "heavier" than SAX. :-)
Might be a somewhat silly analogy, but helps me remember the difference.