When artists use the word "clay," they're referring to a particular combination of earthen particles that tend to be moistened to a point where the homogenized material is plastic and soft. It can be stretched, flattened, mixed, cut, carved, eaten, and generally shaped in a number of ways. After forming the clay into a usable and/or good looking shape, artists dry out the clay (with the aid of a kiln or other heat source) and it transforms into a solid, non-plastic, usable product, often some sort of vessel.
Materials scientists view clay as a colloid of really small particles (from 1/10 of a micron to 50 microns; compare that to a strand of your hair, which is about 100 microns). That means that there are all these teensy hard bits floating around in just enough water to keep them from rubbing too hard against the other hard bits. From studying clays for a long time, these scientists know that most of these clay particles are plate-shaped, hence the name "platelets." They (the particles, not the scientists) are usually pointing in all different directions and barely touch the other particles, except at their corners.
What does this have to do with sintering? What is sintering, anyway?
As the clay dries, water evaporates from between these little flat particles. The space between the particles becomes a little smaller and is replaced with air. As this is going on, the entire ashtray or bowl or statue shrinks a little (how much depends on what kind of clay is used -- kaolin, porcelain, bentonite, earthenware, etc.). However, the particles still retain a small bit of water at their contact points with the other particles.
Enter the sintering! The clay object is tossed in a really hot place (near a fire, in an oven, or in a kiln) and the remaining water evaporates. Thanks to principles of thermodynamics, proximity to a heat source causes the average kinetic energy of the clay's molecules to increase: the clay gets hot. Real hot. So hot that it starts to change form. The corners where the particles touch actually start to melt a little and fuse together. This, reader, is sintering. It's what makes clay the useful material it is.
If you only let your new jar sit in the kiln for a short time (a couple hours), the joining at the edges will only form small bonds. Your jar will hold water, but it will also absorb a lot of it. Its surface will be rough and porous. Think of those big red pots that Home Depot sells for plants.
On the other hand, you can keep your jar in the fire for a very long time (8-24 hours or more). More and more of each particle will melt to the next, causing a stronger bond which fills the nooks 'n' crannies where once there was water. This will generally cause the clay to shrink even more, but it produces a smoother, less porous surface and a sturdier structure. Next time you're eating from Grandmother's China set, flip over a plate and find a spot where there is no glaze -- generally the rim where the plate makes contact with the table.
The plate was high fired, while the flower pot was low fired. The intensity of sintering is a large contributor to the properties of these two products.