There is evidence from the text that Tybalt has written a letter to the Montague household. Benvolio mentions the letter in Act 2, scene 4:
Tybalt, the kinsman of old Capulet,
Hath sent a letter to his father's house.
'His father' is obviously a reference to Romeo's father, since Benvolio and Mercutio are discussing him (Romeo) at this point. The content of the letter is not quite clear, but Mercutio assumes (on his life) that it must be a challenge. Benvolio believes that Romeo would respond to such an invitation.
When Mercutio mentions that,
Any man that can write may answer a letter.
Benvolio answers that it goes further than that - that Romeo would stand up to Tybalt's challenge. Mercutio, however, expresses doubt that Romeo is capable:
... is he a man to
When Benvolio questions Mercutio as to Tybalt's ability, he practically overwhelms him with a diatribe in which he expresses both Tybalt's skill as a swordsman, his status and style, displaying contrasting sentiments towards him. Mercutio clearly both admires and despises Tybalt.
One may therefore assume that the letter Tybalt wrote was a challenge to Romeo, perhaps for attending the Capulet ball uninvited. Tybalt also probably felt hard done by when he was admonished by Capulet for wanting to pick a fight with Romeo when he saw him. He feels that his honour has been tainted and he therefore needs to defend it.