When he was born, she called him Samuel, because I have asked him of the Lord. Oddly enough, "Asked of God" is actually the meaning of the name Saul, not Samuel. Samuel means "Name of God" or "Son of God."
Hannah promised the Lord that There shall no razor come upon his head, indicating that he would be a Nazarite. The original vow is found in Numbers 6. A Nazarite is holy to the lord, separate from the rest of the population. There are different kinds of Nazarites, and anyone who wants to become one may do so for any period lasting at least 30 days. Interestingly, modern Judaism allows fathers but not mothers to declare for their unborn children that he/she will be a Nazarite. A Nazarite has three main restrictions that are not required by non-Nazarites: They may not drink alcohol, they may not cut their hair, they are required to avoid corpses. Lifetime Nazarites, however, are not required to avoid corpses. Modern Judaism allows lifetime Nazarites to cut their hair once per year.
The point of taking a vow of this nature is the same as fasting. Fasting is choosing to abstain from a perfectly allowable behavior in order to remind yourself of an increased devotion to the Lord. We sacrifice something that we are permitted to have or do in order to express the greater value we place on the relationship with the Lord, or our greater desire to earn a specific blessing. A Nazarite was expressing the same desire to strengthen his relationship with the Lord by limiting specific actions in the same way.
Elkenah was an Ephrathite, of mount Ephraim. The books of Samuel don't give any further information about his genealogy other than this opening line. However, he definitely officiated in ordinances at the temple, performing the duties of a priest--duties restricted to Aaron's sons and the Levites. There is a reference to a Shemuel who was the son of Elkenah in Chronicles 6, and that traces back to Levi, but there is some dispute about a) whether this is actually Samuel and b) whether it was written later to try to reconcile the seeming problem with his actual genealogy through the tribe of Ephraim. There's a whole group of people who think he was adopted into the levitical line by Eli, on the basis of the promise his mother made before she conceived him. I'm less inclined to think that. I think that he simply was given the Melchezidick priesthood at some point, obviating the need for him to worry about lineage when he performed ordinances. There are other Old Testament prophets who functioned after this manner, including Elijah, who was of the Tribe of Gad, possibly Isaiah, whose tribe is unknown, and possibly Elisha, whose tribe is also unknown, but who was living where Judah's land was.