Why has Tybalt sent Romeo a letter?
There are two answers to this. The first is that Tybalt is sending a letter to challenge Romeo to a duel. Tybalt is smarting over Romeo's attendance at the Capulet party the night before. Romeo should have known he was unwelcome; by coming to the party he was (in Tybalt's mind) thumbing his nose at the hosts.
"A villain, that is hither come in spite
To scorn at our solemnity this night"
As Susan Woodward notes, Tybalt had wanted to fight Romeo in the heat of the moment, but he had been restrained by his uncle, Lord Capulet. But we, the audience, knew Tybalt wasn't going to drop the matter. He tells us:
"I will withdraw; but this intrusion shall,
Now seeming sweet, convert to bitt'rest gall."
So Tybalt doesn't give up. He simply bides his time until the party is over. Then, the next morning (Act II, scene 4), we learn that Tybalt has sent a letter to "Romeo's father's house." Benvolio relates this to Mercutio, and Mercutio infers—correctly—that Tybalt has issued Romeo a formal challenge to a duel.
This brings us to the second answer to the question, which concerns Tybalt's reasons for sending a letter rather than simply tracking Romeo down in person first. Tybalt is keen on the rather elaborate rules and etiquette of sixteenth-century dueling. As noted by scholar Markku Peltonen, these customs were developed in Renaissance Italy and adopted by many fashionable men in Elizabethan England. People actually read guides and courtesy manuals instructing them on how to behave.
According to William Thomas, an Englishman writing about ten years before the beginning of Queen Elizabeth's reign, issuing written challenges was part of this tradition. If one gentleman insulted or defamed another, it was common for the insulted party to "maketh his defiaunce by a writte called Cartello, and openly chalengeth the defamer to fight..." (cited in the first chapter of Peltonen, see below).
Mercutio tells Benvolio that Tybalt is just this sort of "fashion-monger," a man who styles himself as someone who follows the creed with enthusiasm:
"O, he's the courageous captain of compliments. He fights as you sing pricksong-keeps time, distance, and proportion; rests me his minim rest, one, two, and the third in your bosom! the very butcher of a silk button, a duellist, a duellist! a gentleman of the very first house, of the first and second cause. Ah, the immortal passado! the punto reverse! the hay."
Peltonen, M. 2003. The Duel in Early Modern England: Civility, politeness, and honour. Cambridge University Press.
Tybalt was angry that Romeo and his friends crashed the Capulet party. He originally wanted to confront Romeo at the party, but Lord Capulet told Tybalt to leave Romeo be because he'd just told the Prince that there would be no more fighting between the Capulets and the Montagues. He'd also heard that Romeo was a good guy, so Tybalt would be better off just holding his temper and not start any trouble. Tybalt was seething at being shut down by his uncle, and he took matters into his own hands. The next morning he sent a letter to the Montague house to challenge Romeo to a fight. Benvolio heard about the letter, but Romeo never got it because he never went home after the party. He spent a good part of the night in Juliet's back yard talking to her, and then he went immediately to Friar Lawrence to set up the wedding. By the time he heard that Tybalt wanted to challenge him, Romeo and Juliet were just married. When Romeo refused to fight him, Mercutio (being the hot head he is and always ready to fight) taunted Tybalt until the two of them got into it. Tybalt's challenge left both him and Mercutio dead.