The records from the Nicene Council that survived to the modern day make no mention of setting a canon.
Jerome's preface to Judith, less than 100 years after the council, says that the Nicene Council included Judith (one of the Deuterocanon) among the Sacred Scriptures.
Jerome's Preface to Judith
Among the Hebrews the Book of Judith is found among the Hagiographa, the authority of which toward confirming those which have come into contention is judged less appropriate. [...] But because this book is found by the Nicene Council to have been counted among the number of the Sacred Scriptures, I have acquiesced to your request...
We don't know what Jerome meant by this - doubtlessly there are works and records that existed in Jerome's day that haven't survived to the present day. bradimus' answer gives some good speculation about possibilities here.
Around 900 AD, the Synodicon Vetus was written. This anonymous, psuedo-historical book contains information on synods and ecumenical councils up through the year 887 AD.
On the Council of Nicea, it says that the canon was determined by placing the books on the altar, and after prayer to God the inspired works were found on top.
The Synodicon Vetus [English Translation], chapter 35 (page 29)
The divine and sacred First Ecumenical Council of three hundred and eighteen God-inspired fathers was convened at Nicaea, metropolis of the province of Bithynia. Its presiding leaders were the presbyters Vito and Vicentius taking the place of Rome's Pope Sylvester and his successor Julius, Alexander of Alexandria, Macarius of Jerusalem, Eustathius of Antioch, the presbyter Alexander representing Metrophanes of Constantinople, Hosius the bishop of Cordoba, and Constantine the apostle among Christian emperors. This holy council attached the term "consubstantial" to the Holy Trinity, fixed the time of the divine and mystical Passover, and set forth the divinely inspired teaching of the Creed against all heretics, Arius, Sabellius, Photinus, Paul of Samosata, Manes, Valentinus, Marcion, and their followers. It condemned also Meletius of Thebais, along with those ordained by him, and Eusebius of Nicomedia. The canonical and apocryphal books it distinguished in the following manner: in the house of God the books were placed down by the holy altar; then the council asked the Lord in prayer that the inspired works be found on top and--as in fact happened--the spurious on the bottom.
There are many details only recorded in this book that scholars find spurious, as well as some councils or synods recorded that scholars suspect never happened. This gives credence to doubt the above account of the canon being determined at the Council of Nicea.
Voltaire popularized the story from the Synodicon Vetus around ~1764 AD.
Voltaire's Philosophical Dictionary, Councils, Section III
We have already said, that in the supplement to the Council of Nice it is related that the fathers, being much perplexed to find out which were the authentic and which the apocryphal books of the Old and the New Testament, laid them all upon an altar, and the books which they were to reject fell to the ground.
For legends that the Council of Nicea voted on the canon, the only source I can find that popularized that idea is from Thomas Paine ~1794 AD, though he does not mention the council by name.
Thomas Paine's Age of Reason, Part First, Section 4
Be this as it may, they decided by vote which of the books out of the collection they had made should be the WORD OF GOD, and which should not. They rejected several; they voted others to be doubtful, such as the books called the Apocrypha; and those books which had a majority of votes, were voted to be the word of God. Had they voted otherwise, all the people, since calling themselves Christians, had believed otherwise — for the belief of the one comes from the vote of the other. Who the people were that did all this, we know nothing of; they called themselves by the general name of the Church, and this is all we know of the matter.
In the 1800s, radical Christian Robert Taylor again brought up the Synodicon's theory.
Robert Taylor (~1829), in The diegesis, Appendix, page 432
A.D. 327. Grand Council of Nice in Bythinia, under the presidency of Constantine the Great, gave us the God of God creed used in the communion service. Pappus, in his Synodicon to the council of Nice, asserts, that having promiscuously put all the books under the communion table in a church, they besought the Lord, that the inspired records might get upon the table, while the spurious ones remained underneath, which accordingly happened.
And finally in the modern era, Dan Brown's fictional story implied that Constantine basically chose the canon of the Bible, either at or around the time of the Council of Nicaea.
Dan Brown's fictional Da Vinci Code (~2003 AD)
Constantine needed to strengthen the new Christian tradition, and held a famous ecumenical gathering known as the Council of Nicaea. At this gathering [...] many aspects of Christianity were debated and voted upon – the date of Easter, the role of bishops, the administration of sacraments, and, of course, the divinity of Jesus...
From this sprang the most profound moment in Christian history. Constantine commissioned and financed a new Bible, which omitted those gospels that spoke of Christ's human traits and embellished those gospels that made him Godlike. The earlier gospels were outlawed, gathered up, and burned.
More in depth information available here