I have learned that the word for a person from Britain is a Brit. But, I was browsing the Internet and I saw reading the Guardian that the word is Briton. I researched, and found that Briton is the more accurate word, but I have heard the word Brit used multiple times by people I know over the years.
The following extract from the Grammarist can be helpful:
Briton is the most widely accepted term for people from Britain (which of course is not the same as England and the United Kingdom). Britisher had a brief heyday in the 20th century, but it was always only an American term and was never accepted by Britons themselves. Brit is not offensive, but it is informal.
Of these words, Brit appears most often because it serves as both a noun and an adjective, and probably also because it’s short and not a homophone with Britain. Briton is only a noun, and it only denotes people. Britain is always the correct spelling of the place name.
GDoS shows AmE usage of Brit from early 20th century:
1901 [US] Blackwood’s 169 453: They said; ‘the Brit is at his old game : let us give him time, and smash him [...] as we did at Magersfonteiname .
"Brit" is a word that as an adjective and/or a noun preceded by the indefinite article (a Brit) shows a sharp decline in its frequency of use in the fourth part of the 20th century, while over the same period there is a rise of the plural noun preceded by the definite article (the Brits); that can be read into this ngram.
This word has been used increasingly since the fourth part of the 20th century to name the ethnic group of the Britons as a whole (The Brits) and in the same time period it has not been used much for naming a given individual in that group (ngram), although its usage has steadily increased even if along a comparatively much attenuated slope.
I must say that, personally, as a foreign reader of English (American and British), this is a word of which the mere internalisation - assimilation on the sole basis of context of occurrence without objective reference to the register of use - placed it in my idiolect in a particular category: it stands out as no regular vocabulary item, that is, no plain matter of fact, neutral term, and therefore it has to appear as more or less loaded with connotations. In other words, for the non-native reader that I am, it retains something of the quality of a taboo word for the foreign reader in comparison to words such as "British" and "Briton", as if it were a word that did not belong to the category of "World English", as if it were an "Anglo-Saxon affair".
As a minor supplement to user067531's answer, I note the following instance of "Brits" from a poem titled "What's the News?" in the Indianapolis [Indiana] Journal (February 16, 1900):
O what is the news from Dekel's drift, / Or the news from Slingersfontein? / Is Honey Nest kloof still on the lift, / Or Koodoosberg in line? / Is Majesndie or Jacobsdal / Still held by the Brits or the Boers? / Has Potgieter's been captured yet? / By the men of wiles and lures? / Has Buller quite recover[e]d from / That day at Spion kop— / O tell me! Are the British / Or the Dutchmen still on top?
So the term "Brits" was in use in the United States as slang for "British people" (or in this case "British soldiers") at least as early as the very beginning of the 1900s.
I find it interesting that both this instance and the one from Blackwood's Magazine a year later (cited by Jonathon Green in his Dictionary of Slang and mentioned in user067531's answer) arise in the context of the Boer War—in the latter instance, evidently, being ascribed to a Boer. Could the term have become popularized in the United States out of Dutch South African slang or in dispatches from reporters covering the war?
A slightly earlier instance of "Brits" appears in "The Pious Afrigander," in Crawford Slack, Village Verse Stories and Other Poems (1899):
Sir, it kind of riles me up like en I take a fightin' fit, / When they say the Brits are winnin' en a loosein' all their grit; / Never say a word 'bout fightin' ner don't wait till we get through, / But when we have got our hands full they will tell what they can do.
Slack was a Canadian (seemingly from Ontario), and his poem is written in the voice of a Canadian soldier fighting in the Boer War.
Etymonline's entry for "Briton" includes the following:
c. 1200, "a Celtic native of the British Isles," from Anglo-French Bretun, from Latin Brittonem (nominative Britto, misspelled Brito in MSS) "a member of the tribe of the Britons," from *Britt-os, the Celtic name of the Celtic inhabitants of Britain and southern Scotland before the 5c. Anglo-Saxon invasion drove them into Wales, Cornwall, and a few other corners...
So there is some historical/etymological precedent for using just "Brit", as my American mother often does when referring to my British father's nationals.
In a comment in an SE thread about timezone nomenclature abbreviations, a user self-indentified as "Brit" (emphasis added):
The sort of confusion described in this answer is typical. Most people do not grasp that the three-letter acronyms like PST and AST and IST and so on describe UTC offsets, not geographical timezones. As a Brit, I regularly encounter people who are surprised to learn that the UK is not on GMT the whole year round.