As we begin chapter 24, Abraham begins to realize that Isaac has come of age and needs a wife, and he enlists the assistance of his faithful servant Eleazar to make sure this happens the right way. If you recall from earlier studies, Eleazar was almost like a son to Abraham, and when Abraham was trying to figure out God's covenant to him before Isaac's birth, he was even considering naming Eleazar his heir. Not much is said about Eleazar in the Genesis narrative, but apparently he had served Abraham for a long time, and perhaps came with him from Haran when Abraham first arrived in Canaan. It is even quite possible that Eleazar could have been a second cousin, as he may have even been the son of one of Terah's brothers - again, this is speculating, as Scripture is silent as to the details of Eleazar's life because he only plays a peripheral role in the story. However, given his status with Abraham, what can be certain is that there was a close bond between Abraham and Eleazar, so much that Abraham made him a sort of steward of his estate and trusted his judgment and counsel. That is why it was Eleazar that Abraham charges with the particular task of finding Isaac a suitable mate.
It is at this point the story gets somewhat bizarre for our modern sensitivities, as Abraham demanded from Eleazar a special type of oath in the first ten verses. The oath entailed a ceremony in which Eleazar had to actually lay his hand on Abraham's inner thigh and then swear to carry out what Abraham wanted him to do. This practice, although weird to us in this day and age, was actually a standard oath, and there is much to discuss about this. To begin, an oath like this was always in regard to a family issue. Secondly, the thigh was considered a source of posterity, and in essence what this meant was that the taker of the oath had to swear literally on the testicles of the one he was swearing the oath to (www.gotquestions.org/hand-under-thigh.html - accessed September 2, 2016). There are two reasons for this type of oath in particular, and they are as follows:
1. Seed - Abraham was making his servant swear literally "on his seed" to find Isaac a wife.
2. Circumcision - This was also swearing on the covenant of circumcision as well.