WordWeb Online

1. (used colloquially) overly conceited or arrogant
   "they're snobs--stuck-up and uppity and snot-nosed";
   - bigheaded, persnickety, snooty [informal], snotty [informal], stuck-up [informal], too big for one's breeches [informal], uppish [informal], snobbish
    
2. Dirty with nasal discharge
   "a house full of snot-nosed kids";
   - snotty [informal]

See also: dirty, proud, soiled, toffee-nosed [Brit, informal], unclean