Captain Charles Boycott was a landlord's agent, a man whose job was to collect rents from tenant farmers on an estate in northwest Ireland. At the time, landlords, many of whom were British, were exploiting Irish tenant farmers, and as part of a protest, the farmers on the estate where Boycott worked demanded a reduction in their rents.
Boycott refused their demands, and evicted some tenants. The Irish Land League advocated that people in the area not attack Boycott, but rather use a new tactic: refuse to do business with him at all.
This new form of protest was effective, as Boycott wasn't able to get workers to harvest crops. And by the end of 1880 newspapers in Britain began using the word the way we know it today, not as a person's proper name, but as a tactic of protest.