This refers to the particle size of a powder. A 325 screen has 325 wires-per-inch. The openings between the wires measure 45 microns (half the width of a typical human hair). While 40 mesh (an opening of 425 microns) is fine enough to remove organic material and rocks and coarse sand from a clay), a 325 sieve will remove all contaminants that produce visible specks on firing. It is commonly thought that a 200 mesh sieve is sufficient to de-speck, but that is not so. "200 mesh" is commonly employed is a label attached to air-separated powders, referring to the coarsest particles present (but most of the powder is 325 mesh). A natural clay, screened-as-a-slurry to 325 mesh, will be finer and smoother in the plastic form than even the finest porcelain made from industrial minerals.
Clay particles can be as small as 1/10 of a micron (that is 450 times smaller than 325 mesh), but typically they are in the 1-10 micron range. Thus, they will easily fall through the 325 screen (if not agglomerated). Pretty well 100% of natural quartz and other mineral particles will not pass 325 mesh. Passing clay slurries made from commercial materials through a 325 screen can actually be easier than 200. This is because there is a high population of particles right around 200 mesh, these quickly blind a 200 mesh sieve (jamming themselves in the openings) and reduce its efficiency. Of course, the plus 325 material will need to be discarded, but it is typically a small percentage.
To feel completely smooth-to-the touch clays need to have no particles coarser than 100 mesh.
A 325 screen has 325 wires-per-inch (the finer of the two screen closeups shown here). Those are grains of salt on it (45 micron openings, a typical human hair is 60 microns wide). A 40 mesh screen is much coarser, it has 425 micron openings (that is a particle of quartz trapped in an opening). A minus 45 mesh powder will be too fluffy to drop through a 325 screen. But particles smaller than 45 microns in a slurry will pass. To get a slurry through a screen this fine one needs to take special measures. I needs to have a high water content so it is fluid. Using a soft brush definitely helps. And a source of vibration. And it is necessary to clean the screen often to remove trapped oversize material. 325 mesh screen fabric is fragile and a sieve like this needs to be treated with care. These cost hundreds of dollars.
The coarsest screen is at the top, the finest on the bottom. The opening for each is shown on the label. They are chosen such that each successive screen going down has an opening that is about half the area of the one above it. Using this series you can produce a practical measurement of the distribution of particle sizes in ceramic materials and bodies used in traditional ceramics (structural products industries, like brick, measure coarser particles than this, starting at perhaps 10 mesh and ending at 70). The 325 screen on the bottom is only used sometimes, it is difficult to finer-that-325 particles to pass through it because it blinds. It is not possible to shake powder through sieves that are this fine, samples must be washed through. We use the SIEV test to log results.
To measure particle size in a slurry or powder you need sieves. This is the most popular type used in labs. They are made from brass by a company named Tyler. The range of screen sizes for testing particle size is very wide (obvious here: the top screen has an opening of 56 mm, the bottom one 0.1 mm - the wires are almost too small to see). You can buy these on ebay for a lot less than new ones, search for "tyler sieve". The finer sieves (especially 200) are fragile and easily ripped. It is good to have a 50, 100 and 150.