The War of the Worlds is narrated throughout the book by one person, who mostly tells us his own story. That's a totally normal first-person approach: the main character is the unnamed narrator who tells us all about his adventures during the Martian invasion, even when his adventures are boring. If that were all there was to say about it, we could simply ask, "What does a first-person perspective do for you when you're reading this book?" Maybe it helps us identify more closely with the narrator (since he's telling the story) or maybe it makes his experiences more vivid (because he tells us exactly what he's seeing and feeling).
But there are also some differences between Wells' version of first-person narrator and a more "normal" one. First off, the narrator is a pretty bland character (at least we think so). When he gets upset, we don't usually feel all that upset ourselves. In fact, we think he's a better disaster-story narrator because he's so bland. He doesn't get lost in telling us how he feels, but instead focuses on what crazy things he's seeing: tripods, Heat-Rays, and blood-sucking aliens. That's definitely an interesting change that Wells makes to the usual first-person narrator: he gives us a narrator who is hard to identify with.
Wells makes a second even more serious change to what we might think of as the Standard First-Person Narrator: the narrator tells us about stuff that he didn't see or couldn't possibly know about at the time. The biggest example is the three chapters that are devoted to the adventures of his brother. Not only does he tell us all about what his brother did, he also tells us what his brother thought or felt. For instance, when the brother rescues the Elphinstones from the robbers, the brother "immediately grasped the situation" and "realiz[es] from his antagonist's face that a fight was unavoidable" (1.16.10). There are also other, smaller examples of the narrator getting into other people's heads. For instance, the narrator tells the story of Ogilvy's discovery of the Martian cylinder from his point of view, and includes notes about how Ogilvy makes a "quick mental leap" (1.2.10).
And then there are times when the narrator takes a step back and gives us a broad overview of the situation. This is perhaps most notable at the beginning, with the long introduction about how Martians have been observing humans for a while.
Interestingly, in both the 1953 and the 2005 movie adaptations, a completely different actor gives these God's-eye views of the situation. What does that tell us? It tells us that Wells is a little unusual in having his first-person narrator adopt an occasional third-person perspective.
All that said, Wells does one totally normal thing in that he has the narrator tell his story retrospectively. Not only is it all in the past tense, but the narrator knows how it all turns out. With that advantage, the narrator jumps around in time quite a bit. He often hints at things to come (he looooves foreshadowing) or tells us stuff that he figured out later, as he does when he notes that he didn't know something "at the time," "but later I was to learn" what it was all about (1.15.22).