HaShem had promised Abraham that his progeny would be like the stars of the sky or the sand on the seashore, nearly impossible to count. However, he became an old man, and his only progeny was his son according to the promise - Isaac. He was blessed in very many ways, but still had no grandchildren, and he was almost 140 years old! What to do!
Abraham called for his servant, who isn't named. However, he is described as the oldest of the household, who is also in charge of all that Abraham owned, so it is safe to assume that it is Eliezer. He makes an unusual request of Eliezer: "Please place your hand under my thigh, and I will make you swear by Adonai, the God of heaven and the God of earth, that you shall not take a wife for my son from the daughters of the Canaanites, among whom I dwell. But you shall go to my country and to my relatives, and take a wife for my son Isaac." Genesis 24:2-4.
My first reaction would be "Yuck!" I wouldn't want to put my hand under the thigh of an other man. The word for thigh is jxi (yareich). If you go to Strong's Concordance H3409, you will find that the word jxi can mean thigh or generative parts. This same word is used in Genesis 46:26 and Exodus 1:5, "coming out of the (father's) thigh."
The thigh is also a euphemism for the male genitals. Therefore, the request that Abraham made of Eliezer was even more distasteful to what most of us fellows would be willing to do. However, it was God's will.
"Rashi explains why Abraham chose it (the male genitals) for use in certifying the oath. One who takes an oath must place his hand on some sacred object, such as a Torah scroll or tefillin (phylacteries). Because circumcision was the first precept given to Abraham, and because he fulfilled it through so much pain, it was particularly precious to him, so Abraham asked Eliezer to take his oath upon it. Targum Yonasan renders it similarly. Tanchuma Yashan explains that the reason the Patriarchs cherished the mitzvah of circumcision is that they knew that through it their descendants would be saved from Gehinnom. It is there-fore appropriate to invoke that mitzvah (commandment) before the quest for the woman who together with Isaac would forge the next link in Jewish identity."  Circumcision transformed a profane part of the anatomy into a most sacred part.
Curiously, this concept transfers over into the English language. Our word for testimony has its origins in the word testis, the word for the male testicles. Try to remember this if a judge asks you to give a testimony in court!
I don't see Eliezer protesting about Abraham's method of swearing an oath. (It was later repeated with Joseph requesting a similar oath in Genesis 47:29.) Instead, we see Eliezer showing some doubt about the outcome of him going to Abraham's home country. Eliezer said, "Suppose the woman will not be willing to follow me to this land? Shall I take your son back to the land from where you came?" 
Abraham was also asking Eliezer to do something that was contrary to any personal interest that Eliezer might have. Without a physical heir, Genesis 15:1-2 tells us that Eliezer would then be the heir to Abraham's fortune. By finding a wife for Isaac, Eliezer would no longer be Abraham's heir.
Also, please note that Abraham asks Eliezer to bring back a bride for Isaac from among Abraham's own relatives. Isaac ends up marrying his cousin Rebecca (24:4). Although Torah does not forbid cousins to marry, it may not be wise to do this for many generations. The preferred marriage among Muslims is to marry your first cousin, which has resulted in many mental problems with Muslim cousins marrying each other for the past 1400 years.
Abraham forbade taking Isaac to Abraham's home country, recalling the promise, "To your seed I will give this land."  He didn't want Isaac to have a bride from among the Canaanites. He also told Eliezer that God would send His angel to enable him to find a bride for Isaac. At that time Eliezer agreed to put his hand under Abraham's "thigh" and swear to perform this duty.
Eliezer took with him ten camels and a variety of his master's goods, and set out for the city of Nahor in Mesopotamia. He also had men that accompanied him (24:32), as it would be unsafe traveling long distances alone.
Eliezer came to a well of water outside of the city of Nahor at evening time, "the time when women went out to draw water (24:11)." It was "woman's work" to carry water back then, just as it is in poor countries in Asia and Africa today. The normal time in hot climates for the women to draw water was early morning or just before sunset, to avoid the heat of the day. The "watering hole" was a good place to find a woman! This is where Moshe found his bride, and also where Jacob found his bride (29:6). In Yochanan (John) 4:6-7, this is where Yeshua found the Samaritan woman by the well, but this was at noontime, as she had a poor reputation, and she went when there were no other women at the well, who would not want to be seen with her.
Eliezer made a very specific prayer request: "Behold, I stand by the well of water, and the daughters of the men of the city come out to draw water. And let it come to pass that the damsel to whom I shall say, 'Let down your pitcher, so that I may drink,' and she an-swers, 'Drink, and I will water your camels also' - May she be the one whom Thou hast ap-pointed for Thy ser-vant Isaac."
This wasn't merely an omen or a sign that he was looking for. He was looking for a young lady that was kind and showed compassion to animals - a whole lot of compassion! My Chumash says that each camel can drink 14 gallons of water when thirsty, or a total of 140 gallons for ten camels. She would have to draw all of that water out of a well, and pour it into the trough for the animals to drink from.
Rebecca comes along, and does exactly what he had prayed! This was a very specific answer to a very specific prayer. As it turns out, she was a virgin, and also a relative of Abraham. Eliezer gave her a gold nose ring (24:22; 24:47) weighing a half-shekel, as well as two gold bracelets, each weighing ten shekels. Rashi saw this as very significant. The half-shekel (beka) is the amount that every Jew would contribute to the Sanctuary every year. The two bracelets symbolize the two Tablets of the Law. Their weight of ten shekels each symbolize the Ten Commandments. With today's value of gold, this jewelry would be worth several thousand dollars.
Eliezer requests that he be allowed to lodge in her father's house. She identifies who her family is, and invites him to lodge in her parents' home. Eliezer blessed God once again for answered prayer, and traveled on to Rebecca's home. Rebecca ran on ahead to tell her mother about these things. Her brother Laban heard about these things, and saw the gold jewelry that was given to Rebecca. He then ran to meet Eliezer, and invited him to enter their home. Laban was a wonderful host, unloading the camels, feeding them, and bringing water to wash the feet of Laban and the men who were with him.
Laban's name, by the way, is lavan in Hebrew, which means "white," while most people in this part of the world were (and are) ruddy complexioned. People were often named based on conditions at their birth, or were given names that were prophetically significant. In this case, Laban was very pale, far "whiter" than most around him. Most Jews at that time were "Middle-Eastern" in complexion, much like the Arabs of North Africa and Arabia.
Eliezer explains the reason for his trip, and the amazing answer to prayer. They offer Eliezer generous hospitality. However, Eliezer agrees to only spend the night, and then to return to Abraham, and asks that Rebecca accompany him. She agrees to do so, and they blessed her, saying, "May you, our sister, become thousands of ten thousands (tens of millions), and may your seed possess the gate of those who hate them (24:60)." They then sent Rebecca away with maids.
They arrived at Beer-lahai-roi (Well of the Living One Who Sees Me) in the Negev. Isaac went out to meditate in the evening and saw the camels approaching. Rebecca dismounted from the camel and covered her face with a veil. Isaac then brought Rebecca into his mother Sarah's tent, took Rebecca, and she became his wife (24:67). "He loved her. Thus Isaac was comforted after his mother's death." In earlier times, it wasn't "love and marriage." Often the first time the couple met was on their wedding day. Love came after the marriage. Therefore, it was "marriage and love."
While we are on the discussion of marriage, it should be noted that in Hebrew, it isn't the custom to "say" a prayer. HaShem is much higher than we are. We can speak to other humans because we are on the same plane, or altitude. Instead, the word that is normally used is `Up (nasa). This word means lift or lift up. Curiously, this same word is an acronym for the National Aeronautics and Space Admistration - NASA. What does NASA do? NASA "lifts up" rockets into space! Just a strange coincidence? Perhaps... or perhaps not. Also, `Up (nasa) is the same word used for "marry." When a groom married the bride centuries ago, he would "lift" her up, and carry her to the home he had prepared for her. Today, the men aren't quite so strong (or the bride is too heavy), so we only carry the bride across the "threshold" (the doorway) of the home.
I learned this bit above from Danny Ben-Gigi on Jonathan Bernis' Jewish Voice TV broadcast. Danny Ben-Gigi is traditional, non-Messianic Jew. Although he is very spiritual, he does not embrace Yeshua as his Messiah. However, the analogy he made fits perfectly with Yeshua. The day is coming when Yeshua will `Up (nasa), that is, lift up or marry his bride. If you are "in Yeshua," then you are part of the bride that Yeshua is coming for. He will lift up ("rapture"?) his bride, when there will be the wedding supper of the Lamb. Then He will bring us to the home He has been preparing for us for the past 2000 years.
Although I don't believe in any "Pre-Trib" Rapture, I do believe that it will indeed come as prophesied in 1 Cor. 15:50-54, but only after the anti-Messiah makes his appearance, as prophesied in 2 Thes. 2:1-5. (See also Matthew 24:29-31, Daniel 7:25, and Rev. 13:5.)
 Ishmael was also a son of Abraham, but was not the son of Sarah. The son of promise was Isaac, the son of Sarah.
 Rashi was born in Champagne, France in 1040 CE and died in 1105. He was the first man to write comprehensive commentaries on both Torah and Talmud. He founded the Yeshiva in Worms, Germany, and also earned a living as a wine maker.
 A Targum is a translation of a religious text. In this case, it is an Aramaic translation of the Tanakh (O.T.).
 This entire paragraph is a direct quote from the Interlinear Chumash, p. 120, Schotterstein Edition, Mesorah Publications, Brooklyn (c) 2010.
 Genesis 24:5.
 Gen. 24:7
 Land between the Tigris and Euphrates Rivers.
 Exodus 2:16.
 Torah plus Rabbinic commentary.
 The shekel is a unit of weight, 4/10 ounce.