The characters are nameless and this is purposeful. It illustrates that all humans have a common fate: death.
The stranger speaks of ambition and the glory that awaits him and possibly, the family. This could be seen as referring to the glory of Heaven, but I think he refers to the glory awaiting in life on Earth. A hopeful, wandering dreamer. Ambition is not necessarily a bad thing, nor is it criticized completely in this story. There is, however, a subtle statement that ambition can lead to isolation. While they talk about ambition, the stranger says "it is our nature to desire a monument, be it slate or marble, or a pillar of granite, or a glorious memory in the universal heart of man." It's important to note that he calls it a monument,' rather than a memorial. The father contemplates this, but is content with a slate stone rather than a marble one. (Marble being more modern and prestigious and expected of someone famous.) Even the grandmother worries how she'll look in her coffin. However, the daughter says it is nice enough to sit by the fire and be contented even though they are the only ones thinking of each other.
They are all contemplating death, how they will be remembered and who will remember them. There is a camaraderie in sharing dreams. But it is not ambition that unites them: it is the camaraderie itself. The irony is that they left the safest spot, which was the comfortable anonymity in their home surrounding the hearth.
The young stranger, full of ambition, comes to the door looking melancholy. He is lonely and isolated from the world - until he engages in conversation with the inviting family. His mood changes to cheerful. It's not because he finally has an audience to hear his hopes and dreams (although this is probably what he thinks.) It is because he is in the company of good people. He is not alone. The irony is that what makes him happy is to be in the company of these kind and anonymous people. Yet, he goes on about ambition and fame - seemingly with hope and glory in mind, but it is selfish nonetheless. His isolation, and quest for ambition has always kept him moving, wandering for whatever glory might happen to him. There are many elements of foreshadowing, mostly coming from nature: the wailing winds and so on.
Another quote: "Is not the kindred of a common fate a closer tie than that of birth?" In the end, their common fate, burial under the avalanche, is what they are remembered for. They are not remembered for their ambition or what they achieved, but that they died in anonymity. They have no grave stones or monuments. They are like the sailors who die at sea. But Hawthorne does end by saying "Poets have sung their fate." Not just them particularly, but all who've lead simple, meaningful lives. The smoke billowing from the chimney and the seats still warm around the fire are their grave stones. This scene is a much more warm and vital memorial than a cold stone.