I am so glad to learn that this wonderful poem is still being taught!
The speaker in the poem is an older man trying to seduce a younger woman. This sounds a bit creepy today, but in Marvell's time, "older" would likely have been 30ish, and younger perhaps 15-16. Romance and even marriage between people of such ages was not at all uncommon.
But time does not change human development. A 30-year-old has a great deal more maturity and life experience than does a 15-year-old. The speaker is characterized as someone who knows how fleeting joy can be and is pressed to make the most of each moment. While youth can look on "vast eternity" as stretching ahead to infinity, the speaker's time is short. He is urging his love to seize the day with him, to yield to pleasure (and him) before "Time's winged chariot" hurries too near and the opportunity is passed.