Bilinear and trilinear filtering
Texture filtering deals with how a texture—a 2D image (and other data)—is displayed on a 3D model. A pixel on a 3D model won’t necessarily correspond directly to one pixel on its texture (called a ‘texel’ for clarity), because you can view the model at any distance and angle. So, when we want to know the color of a pixel, we find the point on the texture it corresponds to, take a few samples from nearby texels, and average them. The simplest method of texture filtering is bilinear filtering, and that’s all it does: when a pixel falls between texels, it samples the four nearest texels to find its color.
Introduce mipmapping, and you have a new problem. Say the ground you’re standing on is made of cracked concrete. If you look straight down, you’re seeing a big, detailed concrete texture. But when you look way off into the distance, where this road recedes toward the horizon, it wouldn’t make sense to sample from a high-resolution texture when we’re only seeing a few pixels of road. To improve performance (and prevent aliasing, Austin notes) without losing much or any quality, the game uses a lower-resolution texture—called a mipmap—for distant objects.
When looking down this concrete road, we don’t want to see where one mipmap ends and another begins, because there would be a clear jump in quality. Bilinear filtering doesn’t interpolate between mipmaps, so the jump is visible. This is solved with trilinear filtering, which smooths the transition between mipmaps by taking samples from both.
Trilinear filtering helps, but the ground still looks all blurry. This is why we use anisotropic filtering, which significantly improves texture quality at oblique angles.
To understand why, visualize a square window—a pixel of a 3D model—with a brick wall directly behind it as our texture. Light is shining through the window, creating a square shape on the wall. That's our sample area for this pixel, and it's equal in all directions. With bilinear and trilinear filtering, this is how textures are always sampled.
If the model is also directly in front of us, perpendicular to our view, that's fine, but what if it's tilted away from us? If we're still sampling a square, we're doing it wrong, and everything looks blurry. Imagine now that the brick wall texture has tilted away from the window. The beam of light transforms into a long, skinny trapezoid covering much more vertical space on the texture than horizontal. That's the area we should be sampling for this pixel, and in the form of a rough analogy, this is what anisotropic filtering takes into account. It scales the mipmaps in one direction (like how we tilted our wall) according to the angle we're viewing the 3D object.
This is a difficult concept to grasp, and I have to admit that my analogy does little to explain the actual implementation. If you're interested in more detail, here's Nvidia's explination.
What do the numbers mean?
Anisotropic filtering isn't common in modern settings menus anymore, but where it does appear, it generally comes in 2x, 4x, 8x, and 16x flavors. Nvidia describes these sample rates as referring to the steepness of the angle the filtering will be applied to:
"AF can function with anisotropy levels between 1 (no scaling) and 16, defining the maximum degree which a mipmap can be scaled by, but AF is commonly offered to the user in powers of two: 2x, 4x, 8x, and 16x. The difference between these settings is the maximum angle that AF will filter the texture by. For example: 4x will filter textures at angles twice as steep as 2x, but will still apply standard 2x filtering to textures within the 2x range to optimize performance. There are subjective diminishing returns with the use of higher AF settings because the angles at which they are applied become exponentially rarer."
Anisotropic filtering it's not nearly as much of a hit as anti-aliasing, which is why it rarely appears in menus these days—it's just on, no matter what. Using the BioShock Infinite benchmarking tool, I only saw average FPS drop by 6 between bilinear filtering and 16x anisotropic filtering. That's not much of a difference considering the huge quality increase. High texture quality is pointless with poor filtering.