This chapter continues with the events of chapter 17. I think most conservative scholars consider them descriptions of two separate events (i.e. that the Lord visited Abraham before, talked with him, and then this is a second event that happens later), but the more I read it the more I can see a compatibility between the two, that they could possibly describe the same event because there aren't really any conflicting details.
Anyway, it doesn't make a big difference because the content is essentially the same either way.
So first what we see is the hospitality of Abraham. He rushes to serve the men who draw near to his tent. I think it's arguable to say that Abraham recognized the Lord, and that's why he was so eager to serve these visitors, but a lot of commentators tend to think that Abraham did not recognize them and that this was his normal custom for guests. This is somewhat affirmed later in the bible, in Hebrews 13:2 which says, "Do not neglect to show hospitality to strangers, for by this some have entertained angels without knowing it." While this verse does not mention Abraham, one could argue that it is implicitly referencing the events in Gen 18, and as such would imply that Abraham did not recognize his guests.
This line of reasoning continues with the implication that perhaps the Lord would not have visited Abraham and shared these blessings with him if he had not been openly hospitable, since it more or less says that he approached them and asked them to stay and eat his food. I think this is plausible, but amounts mostly to speculation because we don't actually know what God would have done if Abraham had acted differently.
The heat of the day is naturally the time when rest, shade and water are most appreciated, so this certainly shows Abraham being very courteous.
In verse 9, when they ask about his wife, we start to see these visitors acting in clearly supernatural knowledge as they ask about his wife by name and then restate the promise that we saw in chapter 17, including the specific timeframe of one year. This time, however, Sarah is present and listening (though concealed, so she is overhearing the conversation). Just as Abraham laughed in the last chapter, Sarah laughs in this one, and Sarah and the Lord get into a cute little argument about whether or not she laughed. So her initial response is skepticism. We will see this skepticism vanish when she becomes pregnant.
What is made clear (and what is intended to be made clear) from Abraham and Sarah's responses is that this pregnancy is supernatural. They are both clearly beyond the age of human fertility, and yet they will now have a child by the word of the Lord. This isn't just about chance, this is about God's grace bringing restoration into their lives.
The story transitions then, back to Sodom and Lot. Apparently what we've all been expecting is now about to happen, as the Lord says the time for Sodom's judgment has come.
First thing I see in this is the importance of God letting Abraham into his counsel. There are a couple aspects/implications of this. For one, the basis of Abraham's admission as the text states is his greatness through God's choice, that he is chosen so that God may bless him and his children as they walk in God's ways. But the entire foundation of this blessing is in verse 19, that "his children and his household after him ... keep the way of the LORD by doing righteousness and justice". Another aspect is that of Abraham as the friend of God. Abraham has a very close relationship with God throughout the scriptures, and this is both the basis of God's promise to Abraham as well as God's confidence in him. Lastly, there is the aspect of Abraham as one of the first prophets. Later on Abraham will be called a prophet explicitly, but here we see the pattern for how prophets operate: they walk in closeness with God and from that walk of righteousness (see Gen 15:6) they receive revelation about the plans of God. This is the pattern; prophecy is not magic, it's not divination and it's not telepathy or ESP or whatever. It's walking with God and having God tell you stuff. And that leads me to my next point, what Abraham does with this information.
Abraham's first priority in hearing about the destruction coming to Sodom is to intercede for his nephew. Well, he says "the righteous", but we know that his nephew Lot went to Sodom, so there is that implication here. This is also a very important part of the prophet pattern: intercession. This is something we will see a lot more with later prophets, not as much with Abraham, but it's very important. The biblical pattern for prophecy is really important because modern culture has a very twisted view of what prophecy is, largely because of the actions of a small group of prominent figures who have labeled themselves prophets throughout the past few hundred years of history. Lots of people think of prophecy as some mixture of biblical numerology, end-times eschatology and predicting the future on some obscure or arcane basis. The vast majority of this public perception is completely unfounded from a biblical point of view, but the common vocabulary of calling such people "prophets" results in an unfortunate conflation of the two distinct realms of operation.
I will try to address biblical prophecy more fully at a later date. For now, what I will say is that the core elements are two-fold: closeness in one's relationship with God and intercession on behalf of others, both wicked and righteous, as directed by the prophetic revelation that one receives.
I have already shown some of the behavior patterns from Abraham that reveal his closeness with God and his righteousness, things like his propensity for building altars, his eagerness to serve the strangers, his risktaking to save his nephew from the coalition of four kings, and his willingness to sacrifice everything familiar by leaving Haran and traveling to this unknown land, on the basis of God's promise.
Now in this story we see the second element, which is that after Abraham receives a prophetic revelation (the coming destruction of Sodom and Gomorrah), his first reaction is to demand justice and protection for the righteous. This is different from what we see in many subsequent prophets, where a lot of the emphasis is on mercy for the wicked. The truth is that there is no contradiction between these two appeals and both of them reveal the character of God. Many prophets also demand justice, such as can be easily found in King David's Psalms.
In this case, Abraham's prayer establishes a critical principle in the character of God: that God is not willing to treat the wicked and the righteous alike, that the judge of all the earth will deal justly. This is incredibly important not just because it defines one of God's characteristics, but also because it is so frequently challenged by nonbelievers.
And lastly, God accepts Abraham's appeal: that is, God is (as it appears) willing to change his behavior due to human requests for relief. To a first-time bible reader, this is not necessarily obvious. Why would the omnipotent, world-creating God be willing to listen to a human? To be honest I'm not sure how to answer that, other than to say, well, he does. This is, in some ways, the most meaningful conclusion (and measure) of Abraham's relationship with God. That just like relationships between people, if you ask a stranger to do something he probably won't, but if you ask a friend, he probably will. This willingness to listen on God's part shows that his relationship with Abraham is not (intended to be) strictly a power relationship with God as King and Abraham as the lowly servant. There is an intended parity here, where Abraham can talk to God and God can talk to Abraham and they both listen to each other.
And so we are left with two angels going to investigate Sodom, God's promise of protecting the righteous, and Abraham returning to his home with a renewed promise for his wife to bear a son.