an equiaxed grain

Equiaxed just means equal axes — roughly the same size in every direction, like a pebble or a soap bubble rather than a needle or a pancake. An equiaxed grain is a grain whose width, height and depth are comparable; a microstructure of such grains looks like a foam of roughly round polygons with no preferred long direction.

Grains become equiaxed when they grow into their surroundings at about the same rate in all directions, or when a deformed structure is annealed and fresh grains nucleate and grow without any imposed direction (recrystallization). The tell-tale is that a linear-intercept measurement gives nearly the same mean intercept whichever way you draw the test lines — the structure has no strong directionality. Contrast this with a columnar grain, which is long in one direction because it grew that way, for example inward from a cold mould wall.

Equiaxed grains matter because they usually give the most uniform, isotropic properties — a casting or a wrought part with fine equiaxed grains behaves similarly in all directions, which engineers generally prize. Grain refiners are added to castings precisely to force a fine equiaxed structure instead of coarse columnar grains. Honest note: equiaxed describes grain shape only; it says nothing about crystal orientation. A block of equiaxed grains can still be strongly textured (its grains sharing a preferred crystal direction) and therefore anisotropic despite its uniform-looking shape.