There are professionally purpose-made pottery tools, and also many every-day household objects that can be repurposed for sculpting clay.
I have been collecting and adding to my pottery tool kit over a number of years, and I always find it a joy to take a look at the tools other potters use, ans see what interesting items they have picked up over the years. As a beginner I learned a lot about pottery tools names and uses from asking other potters about what they have in their tool-kits.

Table of Contents
Basic Pottery Toolkit
While every potters toolkit is different, there are a number of essential items that you will find in almost all tool-kits. These basic tools are the essentials you will need to get started.
1. Wire clay cutter

A wire is (as the name suggests) a length of wire usually around 1 foot (30cm) long, and often with wooden toggles attached at each end.
It is used for cutting wet clay, and is an essential part of your toolkit because you will need it any time you wish to remove a finished piece from the wheel head or a bat you have been working on.
It is also needed for cutting while you are wedging the clay (preparing it by removing air bubbles).
2. Sponge

Most potters have a variety of sponges in their toolkits. It is useful to have a few different sizes and shapes as well as different textures.
They are useful for adding and removing water, smoothing surfaces, and general clean-up of your work area. They can also be used for applying glaze if you want a textured appearance, or removing glaze if you .
Many commercially available pottery toolkits include a round palm-sized sponge, but you can use any sponge (cosmetic sponges are popular for fine work).
3. Ribs

Flat tools that come different shapes, and are commonly made from boxwood, though plastic and silicone are also popular materials. Beginners pottery kits commonly include a wooden arc-shaped rib with two flat sides.
Ribs are used for smoothing and shaping on the wheel, and are also useful for scrapping the excess clay off the bat or wheel head.
4. Metal scraper

Similar to ribs, scrapers are flat and (usually) oval shaped, but made of metal, and can have a smooth or a serrated edge. Commercially-available basic pottery kits often include an oval metal scraper.
Scrapers are used for smoothing and cutting the clay, and for making patterns.
5. Wooden modeling tool
Long flat pieces of wood around the length of a pencil, these tools have an angled or pointed end for fine work and getting into tight spaces.
Wooden modeling tools are useful for decorating and trimming extra clay off the bottom of the piece you are working on.

6. Loop tool
Loop tools are shaped cutting tools with a head made from stainless steel ribbon with sharpened edges, and a wooden handle.
Loop tools are used for trimming clay pieces on the wheel once they have reached the leather-hard stage. They can be used to shape and finish surfaces and create functional features of a piece, such as cutting a foot.
7. Ribbon tool

Another cutting tool, ribbon tools are similar to the loop tool, but are often smaller and finer, and double ended with a wooden handle in the center.
The uses of a ribbon tool are similar to those of a loop tool, however because the heads are often smaller they can be used for finer detail work and to get into tighter spaces.
8. Needle tool

The needle tool is a sharp metal needle set in a handle. It is used for cutting, piercing, measuring the depth of the clay, creating patterns, scoring, and finishing fine details. It is a very useful and versatile tool, and will be found in any beginners potter set.
Other useful tools
Edger
Edgers are shaped flat tools usually made of metal or rigid plastic, that are used for finishing the rims of pieces on the wheel.
Clay hole cutters
These are metal tubes which come either with or without a handle. You will need a variety of different sizes as you can only cut one size of hole with each one. You can also get different shaped hole cutters, such as teardrop shaped or heart-shaped.
Clay hole cutters are used for making holes in clay for design and functional purposes such as for a teapot spout, toothbrush holders or holes for hanging pieces.