1) Name of a person with that name.
2) Greek (ζακχαιος=Zakchaios) form of the Hebrew (יךז=Zaccay) word for innocence or purity.
3) Biblical character referenced in Luke 19:1-10

Concerning the Biblical character, we know a few things about Zach and his exploits1 from the reference up there:

A) He was short. Maybe a midget, possibly a smurf, but was definitely short.
B) He was a tax collector. (The chief one in Jericho).
C) He wanted to see Jesus so badly that he climbed a sycamore tree to see Him over a crowd.
D) His community hated his guts.
E) Jesus sought him out as a person with whom to "chow down."
F) After dining with Jesus, he had a change of heart, repented of his jerkiness and repaid everyone he had wronged.

Okay, admittedly, point "D" is an inference, but it seems pretty solid. I mean, he was a (little) big-deal guy in what was a big-deal city back then, so everyone had to know who he was. Further, who he was was the guy that took everyone's money, for "taxes". He pretty much sold his community out to/for king Herod. (i.e. THE MAN, their oppressor, etc...) Now nobody likes taxes, but in Jericho2 taxes and backstabbing little runts who collected them were probably particularly distasteful. Add to that the irony of his name (see def. 2) and it seems pretty clear that he was up for some gut-hatage. That's why having a meal with the guy is a "story" ...nobody wanted to associate with him.

Students of this literature generally find the story to exemplify several key Christian themes:
A)If you seek3 God, you find Him. (from somewhere near the end of Matt. 6 I think.)
B) Jesus came to seek3 and save the "lost" people. (metaphorically, this is the extension of the Hebrew God's offer of a relationship to non-Hebrews and the doctor healing the sick, rather than the healthy).
C) God loves the sinner (person) and hates the sin (behavior). (even if humans think he's a backstabbing little runt)

1 Zach's exploits are also recorded in the song "Zaccheus was a wee little man" which, predictably, has been popular for years among toddlers under protestant rule.
2 Jericho was probably the first city the Israelites captured under the leadership of Josh, and during the time period of this story the Jews were pretty much captives forced to be under Herod's rule.
3 It's interesting to note that the text uses the Greek word "zateo" in both cases, which means not just "seek", but "to seek something you once possessed."