There's nothing particularly magical about the black arrow, but it was an inheritance from Bard's ancestors and (the head at least) may have ultimately been forged in Erebor. This is all we hear about it:
Now [Bard] shot with a great yew bow, till all his arrows but one were spent. The flames were near him. His companions were leaving him. He bent his bow for the last time. Suddenly out of the dark something fluttered to his shoulder. He started—but it was only an old thrush. Unafraid it perched by his ear and it brought him news. Marvelling he found he could understand its tongue, for he was of the race of Dale.
"Wait! Wait!" it said to him. "The moon is rising. Look for the hollow of the left breast as he flies and turns above you!" And while Bard paused in wonder it told him of tidings up in the Mountain and of all that it had heard. Then Bard drew his bow-string to his ear. The dragon was circling back, flying low, and as he came the moon rose above the eastern shore and silvered his great wings.
"Arrow!" said the bowman. "Black arrow! I have saved you to the last. You have never failed me and always I have recovered you. I had you from my father and he from of old. If ever you came from the forges of the true king under the Mountain, go now and speed well!"
The dragon swooped once more lower than ever, and as he turned and dived down his belly glittered white with sparkling fires of gems in the moon—but not in one place. The great bow twanged. The black arrow sped straight from the string, straight for the hollow by the left breast where the foreleg was flung wide. In it smote and vanished, barb, shaft and feather, so fierce was its flight. With a shriek that deafened men, felled trees and split stone, Smaug shot spouting into the air, turned over and crashed down from on high in ruin.
And that's about it. There are no magical qualities, just age (which is always a plus in Tolkien's works) and possible Dwarvish origin; and Bard seems to have been a particularly skillful archer, and to have known about Smaug's one weak spot, which happens to have been right over the heart.