“What is this?” in Korean: 이게 뭐예요? vs. 이건 뭐예요?

이거 뭐예요?
이건 뭐예요?
이게 뭐예요?
These three statements are all basically translated as "What is this?" but with significantly different meanings and tones.

이거 = this, this object, or this one 뭐 = what -예요 = the present tense conjugation of 이다 in polite form
How about the first form? is simply a mixture of these three parts, with no particle being employed. In other words, a particle was missing following 이거. So you can use this sentence to ask "What is this?" in the most impartial manner.

What about the second form, 이건 뭐예요? Is abbreviated from 이거는 뭐예요? or 이것은 뭐예요? Using the subject marking particle -은/는 allows for contrast and a conversational tone. Imagine entering into a room and seeing something on the table that makes you wonder what it is. You ask someone, "이건... 뭐예요?" which means "Um... as for this thing, (waiting for attention), what is it?" Can you say 이건 뭐예요? Can it sound more "soft" and less abrupt or straightforward than 이거 뭐예요? In many circumstances. (However, it all relies on your tone of speech.)

What about the third form, 이게 뭐예요? Is abbreviated from 이것이 뭐예요? Because -이 is the "subject marking" particle, you focus on what this thing is. When you are astonished or outraged by something unexpected, you will likely use 이게 뭐예요. More frequently than 이거 뭐예요? or 이건 뭐예요? (Again, it depends on the circumstances and your tone.)

This is the main distinction between these three sentences, and you will need more practice to fully grasp the subtle nuances of sentence changes like this.