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In computer graphics, texture filtering or texture smoothing is the method used to determine the texture color for a texture mapped pixel, using the colors of nearby texels (pixels of the texture). Mathematically, texture filtering is a type of anti-aliasing (AA), but it filters out high frequencies from the texture fill whereas other AA techniques generally focus on visual edges. Put simply, it allows a texture to be applied at many different shapes, sizes and angles while minimizing blurriness, shimmering and blocking.
There are many methods of texture filtering, which make different trade-offs between computational complexity and image quality.
The need for filtering 
During the texture mapping process, a 'texture lookup' takes place to find out where on the texture each pixel center falls. Since the textured surface may be at an arbitrary distance and orientation relative to the viewer, one pixel does not usually correspond directly to one texel. Some form of filtering has to be applied to determine the best color for the pixel. Insufficient or incorrect filtering will show up in the image as artifacts (errors in the image), such as 'blockiness', jaggies, or shimmering.
There can be different types of correspondence between a pixel and the texel/texels it represents on the screen. These depend on the position of the textured surface relative to the viewer, and different forms of filtering are needed in each case. Given a square texture mapped on to a square surface in the world, at some viewing distance the size of one screen pixel is exactly the same as one texel. Closer than that, the texels are larger than screen pixels, and need to be scaled up appropriately - a process known as texture magnification. Farther away, each texel is smaller than a pixel, and so one pixel covers multiple texels. In this case an appropriate color has to be picked based on the covered texels, via texture minification. Graphics APIs such as OpenGL allow the programmer to set different choices for minification and magnification filters.
Note that even in the case where the pixels and texels are exactly the same size, one pixel will not necessarily match up exactly to one texel - it may be misaligned, and cover parts of up to four neighboring texels. Hence some form of filtering is still required.
Mipmapping is a standard technique used to save some of the filtering work needed during texture minification. During texture magnification, the number of texels that need to be looked up for any pixel is always four or fewer; during minification, however, as the textured polygon moves farther away potentially the entire texture might fall into a single pixel. This would necessitate reading all of its texels and combining their values to correctly determine the pixel color, a prohibitively expensive operation. Mipmapping avoids this by prefiltering the texture and storing it in smaller sizes down to a single pixel. As the textured surface moves farther away, the texture being applied switches to the prefiltered smaller size. Different sizes of the mipmap are referred to as 'levels', with Level 0 being the largest size (used closest to the viewer), and increasing levels used at increasing distances.
Filtering methods 
This section lists the most common texture filtering methods, in increasing order of computational cost and image quality.
Nearest-neighbor interpolation 
Nearest-neighbor interpolation is the fastest and crudest filtering method — it simply uses the color of the texel closest to the pixel center for the pixel color. While fast, this results in a large number of artifacts - texture 'blockiness' during magnification, and aliasing and shimmering during magnification.
Nearest-neighbor with mipmapping 
This method still uses nearest neighbor interpolation, but adds mipmapping — first the nearest mipmap level is chosen according to distance, then the nearest texel center is sampled to get the pixel color. This reduces the aliasing and shimmering significantly, but does not help with blockiness.
Bilinear filtering 
Bilinear filtering is the next step up. In this method the four nearest texels to the pixel center are sampled (at the closest mipmap level), and their colors are combined by weighted average according to distance. This removes the 'blockiness' seen during magnification, as there is now a smooth gradient of color change from one texel to the next, instead of an abrupt jump as the pixel center crosses the texel boundary. Bilinear filtering is almost invariably used with mipmapping; though it can be used without, it would suffer the same aliasing and shimmering problems as its nearest neighbor.
Trilinear filtering 
Trilinear filtering is a remedy to a common artifact seen in mipmapped bilinearly filtered images: an abrupt and very noticeable change in quality at boundaries where the renderer switches from one mipmap level to the next. Trilinear filtering solves this by doing a texture lookup and bilinear filtering on the two closest mipmap levels (one higher and one lower quality), and then linearly interpolating the results. This results in a smooth degradation of texture quality as distance from the viewer increases, rather than a series of sudden drops. Of course, closer than Level 0 there is only one mipmap level available, and the algorithm reverts to bilinear filtering.
Anisotropic filtering 
Anisotropic filtering is the highest quality filtering available in current consumer 3D graphics cards. Simpler, "isotropic" techniques use only square mipmaps which are then interpolated using bi– or trilinear filtering. (Isotropic means same in all directions, and hence is used to describe a system in which all the maps are squares rather than rectangles or other quadrilaterals.)
When a surface is at a high angle relative to the camera, the fill area for a texture will not be approximately square. Consider the common case of a floor in a game: the fill area is far wider than it is tall. In this case, none of the square maps are a good fit. The result is blurriness and/or shimmering, depending on how the fit is chosen. Anisotropic filtering corrects this by sampling the texture as a non-square shape. Some implementations simply use rectangles instead of squares, which are a much better fit than the original square and offer a good approximation.
However, going back to the example of the floor, the fill area is not just compressed vertically, there are also more pixels across the near edge than the far edge. Consequently, more advanced implementations will use trapezoidal maps for an even better approximation (at the expense of greater processing).
In either rectangular or trapezoidal implementations, the filtering produces a map, which is then bi– or trilinearly filtered, using the same filtering algorithms used to filter the square maps of traditional mipmapping.