Translation:I am busy on Tuesday.
I think Jim is spot on (though I'm not sure if it's が or に), but I'll add that you could also specify with something like 今週(こんしゅう)の火曜日(かようび), meaning Tuesday of this week. There's also 先週(せんしゅう、来週(らいしゅう)、and 再来週(さらいしゅう) that I know of, meaning last week, next week (lit. the coming week), and the week after next, respectively.
To add to the rest, は is used here to denote the topic you are saying something about (note that this is not necessarily the one performing the action). In this case you are saying something about Tuesday.
The subject marker が is the thing performing the object. In this case using が would be saying that Tuesday is busy.
In this case は could mean Tuesday is busy. (you are saying something about Tuesday, in this case that it is busy.) but it makes more sense that it means that you are busy on Tuesday (about Tuesday, I am busy. I is implied here)
In real life? It depends on the context of the situation. If someone asked you "What are you doing?" It would be correct to drop the わたしは and just say 「やさいをたべています」(eating vegetables). If the listener can understand whether or not you're talking about yourself based on the rest of the conversation, then it's appropriate to drop it.