What is the significance of the four chariots?
Rashi: They are messengers of Hashem to give authority to the four kingdoms - Bavel, Madai, Yavan and Edom. Radak - a chariot has four horses. This shows the strength of each kingdom in its time - each ruled over the world, and each harmed Yisrael. Yisrael are tested via them.
What is the significance of coming from between mountains of Nechoshes?
Rashi: This shows the strength of these kingdoms. They come from a hard, strong place. Radak - they are strong like mountains. Nechoshes is iron that cuts. We find like this "Holechei Rachil Nechoshes u'Varzel" (Yirmeyahu 6:28). It is stronger than the rock that it cuts and carves. It mentions Nechoshes also to join to Hashem's name - "Nichashti va'Yvaracheni Hashem Biglalecha" (Bereishis 30:27).
Malbim: The two mountains show that they will be divided - two chariots came out from Har ha'Mashchis, and they will harm Yisrael. Two came out from Har ha'Mo'ed to benefit Yisrael. Malchus Bavel was of gold, and Malchus Madai was of silver, like Nebuchadnetzar saw in his dream; both of them passed from the world. Now begins Malchus Yavan, which is compared to copper in his dream; it is after silver.
QUESTIONS ON RASHI
Rashi writes that the chariots represent the four kingdoms. There were already two visions of them (1:8, 2:1)!
Radak: Here they are explained more than above.
Malbim: In the first vision, he saw a man riding on a red horse, i.e. Malchus Yavan, which seized kingship after Daryavesh. Now he saw the kingdoms that will stand in the world until the end of days. Each chariot had four horses, for it is one of the four kingdoms - a different kingdom rules in each of the four directions, and it is influenced from a special Sar that receives from the supreme Merkavah, which is also divided into four directions, like I explained in Yechezkel.