Actually every Shabbat is a “great Shabbat”, but it is a custom to refer to certain ones with a special name. For example, the Shabbat between Rosh Hashana and Yom Kippur is called “Shabbat Shuva” — i.e., “Shabbat of Return” (to God), “Shabbat Nachamu” — the “Shabbat of Consolation”, which follows the fast of Tisha B’Av, “Shabbat Zachor” — the Shabbat before Purim, and others.
You ask why the Shabbat before Passover is called “The Great Shabbat.” That is a “great question” in my opinion! In Hebrew this Shabbat is called “Shabbat HaGadol” and I am aware of the following five reasons for this special name:
1. The Jews in Egypt on the eve of the Exodus were given their first mitzvah (the Torah had not yet been given) which applied to that generation only: "Speak to the entire community of Israel saying, On the tenth of this month, let each one take a lamb for each home...And you shall keep it until the fourteenth day of the month, and the entire congregation of the community of Israel shall slaughter it in the afternoon. And they shall take of the blood and put it on the doorposts and on the lintel...And on this night, they shall eat the meat, roasted over the fire, and unleavened cakes; with bitter herbs they shall eat it" (Ex. 12).
Sefer HaPardes, ascribed to Rashi, explains that the Jews went out of Egypt on a Thursday; therefore the taking of the lamb on the tenth of the month was on Shabbat. The Jews declared "If we sacrifice lambs which are sacred to the Egyptians before their very eyes, surely they will stone us." But God said to them, "Now you will see the wonderful thing I will do for you." Each Jew took his Pesach offering and kept it for four days. When the Egyptians saw this, they wanted to take revenge against the Jews but they were stricken with all kinds of bodily suffering and could not harm them. On account of the miracles that were done on that day, the Shabbat before Passover is known as Shabbat HaGadol.
2. Just as a child who is of the age to keep the mitzvot is called in Hebrew a gadol (an adult), so too the day on which the Jewish People “came of age” and were commanded with their first mitzvah is called “HaGadol”.
3. When the Jews were in Egypt, Moses asked Pharaoh to let them rest on Shabbat. Each week when the Sabbath ended, they returned to their wearisome toil. After the Shabbat when they took the lambs they did not return to their slavery and therefore it was called “Shabbat HaGadol”, the long Sabbath.
4. On this Shabbat large congregations gather to learn the laws of Pesach and this day is therefore called “Shabbat HaGadol” because on it people gathered in large assemblies and learned much about great (important) laws.
5. The haftarah read on this Shabbat is from Prophet Malachi (3:4-24) and speaks of the "great day" of God in the future on which the Messiah will appear.
