"Static" characters are characters that do not go through an important change during a story. A "dynamic" character is a character that does undergo an noticeable change in a story.
There are three characters in Mark Twain's "The Celebrated Jumping Frog of Calaveras County." The first is Jim Smiley. He is the man who Wheeler speaks of, who will bet on anything at all. The second is Simon Wheeler, the storyteller. The narrator (Twain) meets Wheeler in an old bar while Twain is in town searching for the Reverend Leonidas W. Smiley (who Twain later admits may not exist). The third character present, for the story would not be taking place without him, is the narrator, Mark Twain.
Based on the definitions above, none of the characters seem to undergo a major change during the course of the story. Wheeler is a seasoned storyteller who seems quite content to tell his tales to any willing listener. Smiley is no longer living in town.
And Twain only changes in that he decides to leave town without finding the Rev. Smiley. As Wheeler is called away from Twain briefly, Twain prepares to leave. Wheeler returns ready to continue...
“Well, thish-yer Smiley had a yeller one-eyed cow that didn't have no tail, only jest a short stump like a bannanner, and—”
However, Twain has had enough, dismisses the pending topic and narrowly makes his escape.
“Oh! hang Smiley and his afflicted cow!” I muttered, good-naturedly, and bidding the old gentleman good-day, I departed.
Because of the lack of substantive change in the men involved in the story, the characters all seem "static" to me.
A static character in literary terminology is one who undergoes no change in their inner nature relating to values, motives, understanding, or insights etc through the course of the narrative. In contrast, a dynamic character is one who does undergo change in their inner nature relating to the above characteristics through the course of the narrative. When determining whether Twain’s characters are static or dynamic, it is important to note Twain’s purpose in telling this story about Simon Wheeler and his friend, Jim Smiley.
Twain’s purpose is to recount a tall yarn, or folk tale, for the sake of humor and amusement: few (if any) ever claim to have had a personal epiphany upon completion of “The Celebrated Jumping Frog of Calaveras County.” Since Twain’s purpose is to entertain and amuse through an almost ridiculous tale and an equally ridiculous second narrator, there is no reason for characters that develop and have dynamic changes to their inner character (remember the story is a frame story with two narrators: the author of the letter to Mr. Ward and the speaker of the tall tale, Simon Wheeler).
The answer to your question therefore is that both characters are static characters: neither are dynamically developing; neither Simon Wheeler not the author of the letter to Mr. Ward experience any character change. Such character change is called character development--and neither have it--if they were to meet after the time of the narrative, each would be as we leave them after reading the story: tomorrow, just the same as today, static. We can see this static, unchanging quality in the text somewhat through the first narrator:
Dear Sir: -- Well, I called on good-natured, garrulous old Simon Wheeler, and inquired after your friend, Leonidas W. Smiley ... . But, by your leave, I did not think that a continuation of the history of the enterprising vagabond Jim Smiley would be likely to afford me much information concerning the Rev. Leonidas W. Smiley, and so I started away.
We can also see unchanging, static characterization in Simon Wheeler:
At the door I met the sociable Wheeler returning, and he button-holed me and recommenced:
"Well, thish-yer Smiley had a yeller one-eyed cow that didn't have no tail, only jest a short stump like a bannanner, and ...."