Back in the Medieval England, cities had royal charters giving them rights. Someone who had the Freedom of the City could exercise the rights set out in the city charter, which generally meant they could hold property, they could engage in trade, they could elect city officials, they were protected from arbitrary arrest, and so on. Eventually in English history, of course, these rights would be extended to everybody, so now being named to the freedom of a city is pretty much just honorary, like being given the key to the city.
Holders of the Freedom of Dublin City (awarded annually by the Lord Mayor) have several ancient privileges and duties according to wikipedia:
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Holders have the right to bring goods into Dublin through the city gates, without paying customs duties.
Holders have the right to pasture sheep on common ground within the city boundaries. This includes College Green and St Stephen’s Green. (this right was exercised as a publicity stunt by Bono when he brought sheep in to St. Stephens Green).
The right to vote in municipal and parliamentary elections.
Some of the ancient duties are that,
Freemen/women must be ready to defend the city of Dublin from attack
At short notice can be called up to join a city militia
Also a law which was passed in 1454 states that freemen/women must own A bow and a coat of mail, a helmet and a sword.
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As a supplement to Captain Amazing’s answer, it’s worth pointing out that Eastbourne was not incorporated as a borough until 1883 and therefore would never have had any non-honorary freemen.
The borough council’s own website has details of the legal definition of an honorary freeman, as well as a complete list of recipients.