As the literal translation shows, the text simply says after the preparation:
And on the morrow that is after the preparation, were gathered together the chief priests, and the Pharisees, unto Pilate, (Matthew 27:62 YLT)
τῇ δὲ ἐπαύριον ἥτις ἐστὶν μετὰ τὴν παρασκευήν συνήχθησαν οἱ ἀρχιερεῖς καὶ οἱ Φαρισαῖοι πρὸς Πιλᾶτον
Preparation day is an interpretation; the text simply says, the preparation, and as the other answers show, there is no Biblical "Day of Preparation." However, there are certain legal days for which preparation before the required day is either specifically required or logically implied. For example, Tabernacles requires living in booths for 7-days and begins with a day on which work is prohibited. So in order to observe Tabernacles, one's booth must be prepared in advance of the 7-days, and it would be logical to speak of "preparation" during the time leading up to Tabernacles.
With that in mind, here are the potential "preparations" during the time which Jesus was executed:
Weekly Sabbath - no work is permitted
Feast of Unleavened Bread - all leaven must be removed before
Feast of First Fruits - the barley must be harvested
The most likely candidate is preparation for the weekly Sabbath. Since work and shopping were prohibited on the Sabbath, one would need to prepare to observe it. For example during the Exodus the Israelites would gather twice as much manna as on the other days:
On the sixth day, when they prepare what they bring in, it will be twice as much as they gather daily. (Exodus 16:5 ESV)
Additionally, the Resurrection is immediately after the Sabbath and the sequence of preparation, Sabbath, and Resurrection is commonly accepted. This sequence is not without problems. First, it does not equal the three days and three nights which Jonah spent in the belly of the great fish (cf. Matthew 12:38-41). Second, it means the Pharisees went to Pilate on the Sabbath Day. This is possible, but given what the Gospel says about Pharisees and the Sabbath, it seems unlikely they would do this on the Sabbath.
Feast of Unleavened Bread
In addition to eating unleavened bread for 7-days, the Israelites were commanded to remove all leaven from their homes:
Seven days you shall eat unleavened bread. On the first day you shall remove leaven out of your houses, for if anyone eats what is leavened, from the first day until the seventh day, that person shall be cut off from Israel. (Exodus 12:15; also 13:7 and Deuteronomy 16:4))
The command states to remove the leaven on the first day, but like Tabernacles, the first day is a day of no work and logically the removal of leaven would be done before the first day. In fact, the instruction in Deuteronomy is specific: no leaven must be in their presence for 7-days. Modern observation begins with "deep" cleaning of the house weeks before the Passover and embellishes the Passover observance by adding Bedikat Hametz on the evening before Passover.
If by preparation Matthew meant Bedikat Hametz, or a similar extra-Biblical observation, then the sequence would be preparation-Unleavened Bread-Sabbath. This also raises questions but does better align with Jesus' three days/three nights prediction. Additionally, if by "the preparation" Matthew was referring to preparation for Unleavened Bread, it is possible the meaning should not be limited to a single day but the entire process of removing leaven which was finished on a certain day, but may have lasted several days.
Feast of First Fruits
The Feast of First Fruits falls on the day after the Sabbath which follows the Passover:
9 And the LORD spoke to Moses, saying, 10 “Speak to the people of Israel and say to them, When you come into the land that I give you and reap its harvest, you shall bring the sheaf of the firstfruits of your harvest to the priest, 11 and he shall wave the sheaf before the LORD, so that you may be accepted. On the day after the Sabbath the priest shall wave it. 12 And on the day when you wave the sheaf, you shall offer a male lamb a year old without blemish as a burnt offering to the LORD. 13 And the grain offering with it shall be two tenths of an ephah of fine flour mixed with oil, a food offering to the LORD with a pleasing aroma, and the drink offering with it shall be of wine, a fourth of a hin. 14 And you shall eat neither bread nor grain parched or fresh until this same day, until you have brought the offering of your God: it is a statute forever throughout your generations in all your dwellings. (Leviticus 23)
If one considers gathering the items needed to observe this day as work and if they are not gathered on the actual day of First Fruits, then, like the weekly Sabbath preparation before the day would be necessary. While this would seem to be speaking of the day immediately before the Sabbath, the first day of Unleavened Bread is a day of no work, and, depending on the actual custom of observance, it is possible preparation would occur 2-days before the weekly Sabbath: preparation, 1st day of Unleavened Bread, weekly Sabbath, Resurrection. This sequence best fits the Jonah prediction. Regardless of the exact sequence, the Resurrection occurred on the Day of First Fruits.
"Preparation" as used by Matthew is a colloquial term whose precise meaning is uncertain. The issue is compounded by the custom of referring to the 8-days of Passover and Unleavened Bread by a single name: Unleavened Bread (cf. Matthew 26:17; Mark 14:12; Luke 22:7) and/or Passover (cf. Luke 22:1). So even if there were a Biblical "Day of Preparation," there is no way to say it was used in a strict Biblical sense.
1. Kevin Howard and Marvin Rosenthal, The Feasts of the Lord, Thomas Nelson Inc., 1997, p. 67