And Hashem appeared to him (Bereishis 18:1)

This occured in the 2047th year since Creation. It was three days since Avrohom had performed his own Bris Milah and it was Erev Pesach.

--Seder Hadoros

And Hashem appeared to him (Bereishis 18:1)

Rashi comments, "(Hashem appeared) to visit the sick". The words "Hashem appeared" prove that He only came to visit and not speak because Hashem's appearance is not accompanied by any speech or conversation with Avrohom. We can learn Derech Eretz (proper behavior) from here. A person should visit the sick even if he can't speak to that person, perhaps the sick person will be sleeping when he arrives. The visit will still please the sick person when others tell him that someone came to visit.

--The Rosh, quoted in Shaarei Aharon

...and lean yourselves under this tree (Bereishis 18:4)

Wherever Avrohom Avinu lived in Eretz Yisroel, he planted trees for shade and beauty. Through the tree that his guests would sit under, he knew if they believed in Hashem or not. When the tree branches spread over the guest and shaded him, Avrohom knew that this guest believed in Hashem. If the branches raised themselves upward and didn't provide shade for the guest, it was a sign that he did not believe. In such a situation, Avrohom Avinu would exhort the guest until he believed.

--The Zohar HaKodosh

And he said, "My masters, please, may I now find favor in your eyes. Please do not pass from your servant." (Bereishis 18:3).

It is proper for one who does something good for others to appear as if they have done him a favor by accepting his kind gesture. He should not become exalted in his own eyes for the favor he has done them. We find Avrohom showed these visitors that they gave him a gift by coming to his house and accepting his hospitality.

--The Sh'loh HaKodosh

For I know him (Bereishis 18:19).

You need to understand that Hashem's Providence over man in this world is general and specific. For other living creatures it is only general, not specific. His Providence over them is only to allow perpetuation of the species. The specific Providence over man is divided into two categories. He watches over man's thoughts and actions. He also watches over man to protect him and save him from harm. Watching over man's actions applies to all people, Jew and gentile. The Divine Protection is only for the righteous of Yisroel. HaKodosh Boruch Hu saves the Tzadikkim from the things in which others are allowed to stumble. He does not forsake His pious ones. He does not remove His Eye from them. His Divine Providence watches over the Tzaddik constantly and never separates from him at all.

--Rabbeinu Bechaye

Please take your son, your only son...(Bereishis 22:2).

Since Hashem had promised Avrohom that, "In Yitzchok your seed will be called", He spoke to him now with words of request and not command. HaKodosh Boruch Hu requested that Avrohom release Him from His promise of giving him a son. He asked Avrohom to offer his son as a sacrifice. This test is a great praise to Avrohom. He wasn't commanded to do it. Hashem desired him to do it. He wanted Avrohom to perform the Akeidah. With his great love of fulfilling Hashem's Will, Avrohom went to sacrifice his son in joy. The posuk says, "your only son". Avrohom certainly believed that Hashem would give him another son, just like He gave him this one. Hashem told him, however, that he would not have another son. Yitzchok would be his only son with Soroh. This only son was the one to be offered on the Mizbeach (altar). Even though this meant that all of Yisroel would be sacrificed through this Akeidah, Avrohom accepted it upon himself with Simcha (joy).

--The S'fas Emes