The narrator of The Strange Case of Dr. Jekyll and Mr. Hyde is an anonymous third person limited narrator who directs how the story will be told, beginning in Chapter 1 where thenarrator says:
"Mr. Utterson the lawyer was a man of a rugged countenance that was never lighted by a smile; cold, scanty and embarrassed in discourse..."
This narrator tells the story through the experience, thoughts, feelings, actions, motives of Mr. Utterson, which is why a story about Jekyll and Hyde starts out with a description of Mr. Utterson. Since the narrator is limited and not omniscient, (1) Stevenson always orients the story from Utterson's point of view and (2) Stevenson was free to expand his narratorial options by having three different people, therefore three different voices, take over the narration at various points in the story.
The first place in which another narratorial voice takes over the story is in Chapter 1 in which Mr. Enfield, Utterson's distant cousin and confidant, introduces Mr. Hyde by telling Utterson about a most peculiar incident that he was involved in that centered on Hyde. So in this instance, Stevenson employs the literary technique of an embedded narrator: a third person narrator telling about a character narrating a story to another character, a technique Joseph Conrad also used in Heart of Darkness.
In Chapters 9 and 10, Stevenson employs another technique to vary the narratorial voice although the narrator remains the same third person limited narrator, as is confirmed in Chapter 8: " Mr. Utterson was sitting by his fireside one evening after dinner,...." In chapters 9 and 10, Stevenson employs two letters, one from Dr. Lanyon to Mr. Utterson and one from Dr. Jekyll to Mr. Utterson, to continue the story of Dr. Jekyll and Mr. Hyde. This is called an epistolary (letter) technique.
So, while the narrator is an invovled ("Never (she used to say, with streaming tears, when she narrated that experience), never had she felt...") though objective third person limited (one point of view through one character) narrator, the narratorial voice varies through embedded narration (Mr. Enfield) and two instances of epistolary narration in which Dr. Lanyon speaks (Chapter 9) and then Dr. Jekyll speaks (Chapter 10).