From Diablo Wiki
Attack Rating, often abbreviated AR is an integral stat relating to the combat in Diablo II.
Attack rating is used to calculate how likely physical attacks are to hit their targets. AR is boosted by numerous item bonuses, charms, and skills, as well by raising the dexterity stat. Other factors, including Clvl, Mlvl, and defense, are also checked when the game calculates whether or not an attack will hit.
Generally speaking, higher Attack Rating gives your character a better chance to hit the target. The formulas for determining to/hit have changed numerous times in D2C and D2X patches, so how important AR is has risen and fallen in different versions of the game. Since v1.10 AR has become much more important, and combat characters now aim for very high AR, 10,000 or more, to ensure at least 85% to/hit against high level monsters.
AR suffers considerable diminishing returns as it rises, and adding thousands more AR might actually add only 1 or 2% to your to/hit, if it's very high to begin with.