I understand there are the ten commandments, which I naively presumed were the only "commandments" in the bible. I am under the impression that this is not true and that there are additional commandments. If this is true, what defines an instruction as a commandment in the bible?
A Biblical commandment as applied to scripture is a directive from God to a people group. In the Old Testament, this was always from God the Father and was always directed to Israel by way of the Prophets. Most of the commandments are found in the Pentateuch, although God spoke through other prophets during different times in Israel's history.
In the New Testament, commandments came from the Messiah directly and were directed to Christian believers. The Apostles did not add to the commandments of Messiah but referred to Messianic commandments in their writings to the churches.
Any other commandments found in scripture would not be considered Biblical commandments but instead represent historical commandments from humans to other humans.
A 'commandment' in the Bible is exactly what the definition says: "a command or mandate.". The Bible records many commandments. Many of the are ordinary commands made by people to others, such as kings or rulers commanding their subjects to do specific things. However when we walk about 'biblical commandments' it usually refers to God's commands to his people, and that's what the rest of this answer will talk about.
As you have correctly deduced, there are more than 10 commandments in the Bible. Orthodox Judaism identifies 613 commandments from God, in what Christians call the Old Testament. Christianity isn't going to disagree seriously with such a statement, though according to Christian theology they are not all still applicable. In the New Testament there is no agreement on how many commands there are, but there are different teachers who identify anything up to 1050. This page gives an overview of some numbers that have been attached to them. For Christianity the commands of Jesus have the full force of commands by God.
Given that there are so many commands, you might wonder why the Ten Commandments are so often talked about. The Ten occupy a special place in that they were the commands of God written on the tablets of stone and brought down the mountain by Moses at the time of God's first formal covenant with the Jewish people. They have always been considered as a 'foundational statement' of Judaism and Jewish Law, and by extension of Christianity, as well as a good summary of the larger numbers of commandments.