Chance to block a close-combat attack fends off melee attacks from monsters, Chance to avoid Arrows their arrow and magic attacks. Both bonuses follow the same maths - everything on this page applies to both alike. You can tell a melee attack was blocked by the Block text that appears, and that you dodged a ranged attack by the Dodge text.
On the old continent and in the Abandoned Fortress the maths is simple: your block or dodge rate equals your bonus value. With a bonus of 40 you fend off 40 % of attacks, with 100 every single one. This does not apply to skill attacks.
From Yohara onwards this no longer holds. There the monsters carry the hidden stat SungMaHt, which pushes your block and dodge rate down. The higher it is, the less you fend off.
applies for SungMaHt > 0
How well is this backed up?
The formula comes from our own measurement series with more than 32,000 counted attacks across 28 combinations of monster and bonus value (chi-square 32.8 at 28 degrees of freedom, p = 0.24). It contains no fitted parameter.