With Simcha Felder resigning from the City Council for a job with John Liu, a series of elections are needed to fill the seat. The first one will be for a term that expires at the end of the year; the next will be for the remainder of the four-year term Felder was just elected to.
Election law expert Jerry Goldfeder at Stroock & Stroock & Lavan LLP, explains it all in an email to me:
Within 3 days from the vacancy the mayor issues a proclamation for a
special election for the city council seat, which is held between 45 and
55 days after the vacancy. The person elected at the special serves
until end of year. There is a primary and general election later this
year to fill the vacancy for the rest of the term.
Petitioning starts on the day of the proclamation–so candidates need to
start to get ready.
There are no primaries for the special election.