In the time of Tar-Ancalimon's father Tar-Atanamir, and of his grandfather Tar-Ciryatan, the Númenóreans had begun to turn against the Valar and the Eldar, and take pride in their own strength and wealth. Tar-Ancalimon continued down this path, and the histories of Númenor record that the division of the Númenóreans began in earnest during his reign.
Politically, the realm became split between two parties, the King's Men, who followed Tar-Ancalimon's policies, and the Elendili, who favoured continued friendship with the Eldar. For the next millennium, Númenor's history was bound up with the dissent between these two parties, and ultimately the King's Men were to help bring about its destruction.
That lay in the far future, though: during his reign, Tar-Ancalimon turned away from the traditions of the past. He discontinued the dedication to Eru, which had taken place at the Meneltarma in the middle of the island. The tradition of the taking of the Kings' names in Elvish he maintained, though, partly through fear of ill-fortune. This is perhaps because of a dark omen of Tar-Ancalimon's reign: the Eagles of Manwë, who had nested in the tower of the King's palace at Armenelos since Númenor's founding, left their eyrie and never returned.
Tar-Ancalimon died as he had lived, clinging to the last vestiges of life as his father had before him. He was succeeded by his son Tar-Telemmaitë, a man greedy for riches and silver, who followed in all the ways of his father.
...or, ruled 165 years from II 2221. Tar-Ancalimon's date of accession is a thorny problem. The detailed account of The Line of Elros (in Unfinished Tales of Númenor and Middle-earth) gives a definite accession date of II 2221, while the original form of the Tale of Years for the Second Age (in The Peoples of Middle-earth) includes an entry '2060-2251 Reign of Tar-Atanamir...' (Tar-Ancalimon's father). We follow the dating from the original Tale of Years text here.
To complicate matters yet further, the Tale of Years that appeared in earlier editions of The Lord of the Rings included the entry '2251Tar-Atanamir takes the Sceptre'. This is in significant conflict with all other sources, and more recent editions have been amended to read '2251 Death of Tar-Atanamir. Tar-Ancalimon takes the Sceptre'.
'Bright' is here meant in the sense 'radiant', 'brilliant'. Compare the famous Elvish phrase Aiya Eärendil Elenion Ancalima (for example, from The Two Towers IV 9, Shelob's Lair), which can be translated 'Behold Eärendil, brightest of stars'.