Hand-building

Handbuilding is exactly what it sounds like; using your hands to form an object out of clay. It encompasses some other, more specific, forming methods as well, like coil building and slab building, and is often used in conjunction with other forming techniques. Ceramic sculpture would fall into the hand-building category since it incorporates many different techniques.  

Slab Building

A process whereby slabs of clay are rolled or pounded out, either by hand, with a slab roller or rolling pin, and then used to construct objects or vessels. Depending on the aesthetic or design requirements, clay slabs can be used from the wet stage up through leatherhard. Many slab builders embellish the slabs with surface designs or textures prior to construction because it is easer to create surfaces when flat rather than after a piece is made.

Check out these short videos on slab building provided by Ceramic Arts Daily.
Slab Vase Project
Slab Tray Project

Coiling

This is perhaps the most simply understood technique for making clay vessels and sculptures. Coils of clay are rolled out, and are built up in a spiral fashion, with the coil being added joined to the coil below it layer after layer until the desired wall height and profile is achieved. This is a great entry-level technique, and will teach you a lot about what clay will and won’t do when wet.

Throwing

Wheel throwing is probably what most people think of when they think of making pottery. A potter will use a manual or electric potters wheel to center a ball of spinning clay, open it into a vessel, and lift and shape the walls while the wheel is spinning, thereby producing a symmetrical vessel. These vessels can then be textured, decorated, reformed into alternative shapes, or adorned with handles for cups spouts for tea pots or anything your imagination can conjure up.
Check out this video provided by Ceramic Arts Daily on throwing.
Faceted Bowl 

Extruding

This technique was borrowed from industry, and has been adapted by clay artists to fit the studio scale. An extruder is a tool with a plunger that forces wet clay through a tube and out through a die at the end of the tube, effectively squeezing the clay into a shape that is determined by the die. Dies can be purchased or made, and the possibilities are many. They are often used to form shapes that are attached to items that have been constructed through other techniques (like handles for mugs).

Slip Casting

Slip Casting involves using specially designed molds that you pour liquified clay into to form the internal shape of the mold. The clay is poured into the mold and allowed to sit for a while so the mold can wick some of the water from the clay, causing a thin layer of more rigid clay to take the shape of the mold. The excess slip is then poured out of the mold and set aside to dry. When it has sat for enough time, the mold is opened and the hollow clay form is removed. Once removed the seams from the mold are scraped smooth, and the piece is allowed to dry further until it is ready to be bisque fired.

Check out this video supplied by Ceramic Art Daily on Slip Casting
One Piece Slip Mold