A peasant is a poor farmer of low social status who owns or rents a small piece of land for cultivation (chiefly in historical use or with reference to subsistence farming in poorer countries).
• used informally, the word means "an ignorant, rude, or unsophisticated person; a person of low social status."
A farmer is a person who owns or manages a farm.
Definition for peasant - Oxford Dictionaries Online (World English)
Definition for farmer - Oxford Dictionaries Online (World English)