notadictionary.com
J'ai failli m'étouffer avec ma salive
→ “I almost choked on my saliva”
J'ai — I have → here, 'j'ai' is the contraction of 'je' (I) + 'ai' (have); used as an auxiliary verb to form the past tense. In this context, it means 'I nearly...'
failli — nearly / almost → indicates something that almost happened but didn't; from the verb 'faillir'
m'étouffer — to choke myself → reflexive verb 's'étouffer'; 'm'' is the reflexive pronoun meaning 'myself', and 'étouffer' means 'to choke'. The 'é' shows the stress in pronunciation.
avec — with → preposition indicating accompaniment or instrumentality
ma — my → possessive adjective agreeing with 'salive' (feminine singular)
salive — saliva → noun referring to the liquid produced in the mouth