The phrase means, "enjoy the moment," or "enjoy the present," or "seize today’s opportunities." Most literally, however, it means, "seize the day." The "carpe" part of the term comes from "carpo" meaning "to pluck" or "to pluck off." It’s related to the Greek "karpos," which means fruit (Karpos is the New Testament word used for fruit, particularly for spiritual fruit when combined with "tou pneumatos," that is "karpos tou pneumatos" = "fruit of the spirit" (cf. Galatians 5:5

The diem part of the phrase means "day." Perhaps when you’ve gone a business trip you were concerned with per diem, which referred to the allowance granted to you per day for your daily expenditures. Diem is related to the Latin "deus" meaning "god," from which came Zeus, who in Greek mythology was the supreme deity and ruler of the celestial realm. Therefore, the god who controlled the cycle of days became known as Zeus, a name that probably ultimately derived from the Latin word for "day."
The idea of being concerned with time and living every moment to the fullest is a biblical concept as well. The Scriptures teach us to "number our days" (Psalm 90:12


Note: As an aside, a similar expression to carpe diem is in diem vivere, or "to live for the day." The latter phrase can be found used in the Latin works of Cicero.