Highly humble and literary. A father uses 犬子 ('dog son') to refer to his own son when speaking to others, downplaying him. Pair with 令郎 (lìngláng, 'your esteemed son') when referring to the listener's son. Sounds archaic in casual modern speech.
history
Said to originate with the Han dynasty writer Sima Xiangru (司马相如), whose childhood nickname was 犬子. Later generations adopted the term as a humble way to refer to one's own son.