The "mane" of a horse refers to the hair that grows on the top of its neck. The immediate context of vv. 19–25 is imagery associated with this war horse that fearlessly and eagerly carries its rider into battle. Ultimately God is questioning Job whether he is the one who gives horses their strength and power (the rhetorical answer is an emphatic "No!").
A challenge with this translation is that רַעְמָה (ra'mah) is a hapax legomenon. The word is often connected with its (likely) root meaning "thunder" (רַעַם, which may also carry the connotation of "quaking" or "rumbling," as these are also associated with thunder).
The Septuagint translators understood it in the sense of fear or terror ("ἐνέδυσας δὲ τραχήλῳ αὐτοῦ φόβον" == "[have you] clothed its neck with terror" || "put fear on its neck"), although this may have to do with the root meaning of the Greek word for "mane" being "fear." This seems to fit the context well. The Latin Vulgate focused on the sound ("neighing").
A. B. Davidson thought it referred to the quivering of the neck rather than the mane. Gray thought the sound and not the movement was the point. But without better evidence, a reading that has “quivering mane” may not be far off the mark. But it may be simplest to translate it “mane” and assume that the idea of “quivering” is part of the meaning.
The quivering mane may imply an allusion/pattern as Job 39:13 references the "vibrating wing" of the ostrich. Assuming the connection with the root word for "thunder," the poetic implication of v.19 might be:
poetically for, "he with arched neck inspires fear as thunder does." Translate, "majesty...."
Translators essentially agree that "the image is one of magnificent power." Individual translations might convey this differently (whether more literally: "flowing mane," or by attempting to retain the metaphorical language: "clothed with thunder"), but that is the overarching imagery being conveyed.