How were the Vatican I and Vatican II councils related? What was the difference between The First Vatican Council and the Second? Was the Second Council a continuation of the first? What does a 'Pastoral' Council mean?
The First Vatican Council (8 December 1869 to 20 October 1870) defined dogmas on ecclesiology (incl. papal infallibility) and faith and reason in its two documents

Pastor Æternus (First Dogmatic Constitution on the Church of Christ)
Dei Filius (Dogmatic Constitution on the Catholic Faith),

respectively. The council was cut short due to the outbreak of the Franco-Prussian war and invasion of Rome.
In 1923, Pope Pius XI considered it dangerous to reconvene the council because—as Cdl. Billot advised him, contrary to the counsel Pius XI sought from almost 40 other theologians—(from The Second Vatican Council: An Unwritten Story II 4. §b)

it would impossible to conceal the existence of profound differences, within the episcopacy itself, on social, political and economic questions, and their relations with morality and the rule of faith. Because of their complexity, these questions will be presented under different aspects depending on the country, and they will give rise to discussion that will run the risk of being extended and prolonged indefinitely.
So here we have the most important reason that would seem to me to militate absolutely against the idea. The resumption of the council is desired by the worst enemies of the Church, that is, by the modernists, who are already prepared—as the most certain information testifies—to take advantage of the general situations in the Church in order to start a revolution, the new French Revolution of their hopes and dreams. It is useless to say that they will not succeed, but we will relive the sorrowful days of the end of the reign of Leo XIII and the beginning of the pontificate of Pius X; we will see even worse, and it will be the total destruction of the happy fruits of the encylical Pascendi which had reduced them [i.e. the modernists] to silence.

Here is what Cdl. Ottaviani said about John XXIII's idea to convene Vatican II (ibid. §c):

He had spoken about it to me from the moment of his election. Or rather, to be more precise, it was I who visited him in his little room at the conclave on the eve of the election. Among other things I told him, “Your Eminence, it is necessary to think about a council.” Cardinal Ruffini, who was present at the conversation, was of the same mind. Cardinal Roncalli adopted this idea and later had this to say: “I have thought of a council from the moment when I became pope.” It’s true, he welcomed our suggestion.”

The nebulous term &quot;pastoral&quot; was used to describe how Vatican II refused to condemn errors and heresies.
For more info on Vatican II and how it differed from all the previous ecumenical councils, see the Little Catechism on the Second Vatican Council:

Part One

Part Two

Part Three

Part Four - The Four Constitutions: Lumen Gentium on the Church

Part Five - The Four Constitutions (continued):  Dei Verbum (on sources of Revelation)

Part Six - The Four Constitutions (continued):  Constitution Sacrosanctum Concilium on the liturgy

Part Seven - The Four Constitutions (continued): Gaudium et Spes - The Place of the Church in the Modern World

Part Eight – The Nine Decrees (decrees 1-5)

Part Nine - The Nine Decrees (decrees 6-9)

Part Ten – The Three Declarations (first of three: Dignitatis Humanæ: on religious liberty)

Part Eleven – The Three Declarations (second and third of three): ‘Nostra ætate’ and ‘Gravissimum educationis’

Part Twelve – CONCLUSION