Vatican Council I was the first Ecumenical Council to be held at the Vatican in Vatican City which is an independent country.
It was held in Vatican City from 1962-1965.
The council was held mostly at the Vatican but some meetings were held at other venues in and around Rome.
The First Vatican Council was held in Rome, Italy in the Basilica of St. John Lateran.
It was held 1869-70.
In Vatican City, thus the name Vatican Council II.
Vatican II was held in four sessions in the autumn of the years 1962-1965.
There were exactly no doctrines defined at the Second Vatican Council. The Second Vatican Council was the first purely pastoral council ever held in the Church.
The First Vatican Council was convoked by Pope Pius IX on 29 June 1868 and was formally closed in 1960 prior to Vatican II
The Second Vatican Council was held at the Vatican, obviously, Ecumenical Council are named after the location where they are held, this was the Second Ecumenical Council to be held at the Vatican. It was convened in 1962 by Blessed Pope John XXIII, of happy memory; it was closed by Pope Paul VI, of happy memory, on December 8, 1965. It's purpose, according to Pope John XXIII was to "update the Church" to bring its presentation of the faith up to modern standards in order to better evangelize the world, and to call individual Christians to a deeper holiness.
Most receive their name from the location where the council is held, such as the council of Trent, Vatican Council, etc.
This council is generally called the Second Vatican Council or simply Vatican II. It is a council that was held for Roman Catholics, but it is not accepted or recognized by the Eastern Orthodox Church.