When G-d wanted to speak to Moshe for the first time, He grabbed Moshe's attention by showing him a burning bush that wasn't being consumed by the fire. But why this? From all the possible things to impress Moshe, He chose exactly a burning bush. There must be a reason for why He chose this method. A meaning behind it.
The midrash in Sh'mot Rabbah (2:5) offers some explanations:
Why fire? To inspire him with courage so that when he comes to Sinai later he is not afraid of the fire.1
Why a thornbush? R' Yehoshua b. Karchah said: to teach that no place is devoid of God's presence, not even a thornbush.
R' Eliezer said: just as the thornbush is the lowliest of all trees in the world, so too Israel was lowly and humble in Mitzrayim.
R' Yose said: just as the thornbush is the prickliest of all trees and a bird that goes into it does not come out unscathed, so was the slavery in Egypt more grevious before God than all other servitudes in the world.
R' Yochanan said: just as the thornbush grows near any water, Israel grew only in virtue of Torah which is called water.
R' Pinchas b. Hama ha-kohein said: just as when a man puts his hand into a thorbush he doesn't feel it at first but when he takes it out he is scratched, so too when Israel went into Mitzrayim nobody really perceived them, but when they went out they did so with signs and wonders and battle.
R' Nachman b. R' Shmuel b. Nachman said: some trees produce one leaf,2 some two or three, but the thornbush has five. God said to Moshe: "Israel will be redeemed for the sake of Avraham, Yitzchak, Yaakov, and for your sake and the sake of Aharon".
Not attributed: the heavenly fire shoots out branches upward, burns but does not consume, and is black in color, while fire used here below does not branch upwards and is red and consumes. Why show Moshe this? Because Moshe had thought that the Egyptians might consume Israel. So God showed him this and said: "just as the thornbush is burning and is not consumed, so the Egyptians will not be able to destroy Israel".
Also not attributed: because when God spoke to Moshe, Moshe was at first unwilling to desist from his work, so God therefore showed him this so Moshe would turn his face to see and speak with Him.
I have also read divrei torah that point out that it took a fair bit of attention to notice: noticing a fire is easy, but Moshe had to (a) decide it was important (it's out in the wilderness, so it's not like it's a danger) and (b) look at it long enough to notice that it's not being consumed. Like running after the single sheep from the flock, this shows an attention to small matters that many would ignore. So it might have been a test of sorts. I don't remember a source for this, though.
1 The burning bush was actually on Horeb (Sh'mot 3:1), which is also Mount Sinai.
2 It says leaves, but a tree with a single leaf seems unlikely and I wonder if it's talking about the number of points on its leaves. I'm not a botanist.
Rashi says God appeared in a thornbush to show He was sharing the Jews' pain.