Though the name may be daunting, this one is actually an easy concept once chopped up into littler pieces.
The third-person part of things means the narrator isn't a part of the story. He's an outside entity looking in. We can tell this because the narrator uses the third-person pronouns "he" and "she" to describe every character. If the narrator were inside the story—what's called a first-person narrator—there'd be an "I" pronoun popping up. Since we don't have one, we have to go with a third-person type narrator.
Now for "omniscient." That part means the narrator can go anywhere and know anything. The narrator isn't tied down to the perspective of one character at all times (like, say, Harry Potter and the Sorcerer's Stone). He's all-knowing, like the god of the story. In fact, "all-knowing" is literally what omni ("all") scient (knowing) means.
We can tell this is the case for "Rikki-Tikki-Tavi" because the narrator occupies the minds of many different characters. Usually, the narrator focuses his storytelling efforts on Rikki-tikki's point of view. But every now and then, the narrator will jump to another character's perspective. For example:
Nag was thinking to himself, and watching the least little movement in the grass behind Rikki-tikki. He knew that mongooses in the garden meant death sooner or later for him and his family; […] (27)
Notice how this paragraph has leapt out of Rikki-tikki's perspective. We're receiving Nag's thoughts and looking at something behind Rikki-tikki's back. These two bits of information can't be known to Rikki-tikki, so the information must be coming from a source outside the story.
This type of back and forth between character's thoughts and perspectives is a clear sign we've got an omniscient narrator. Added to the third-person we discussed earlier, and we get "third-person omniscient" narrator. Simple enough, right?