Frying pan, fry pan or frypan, also skillet, flat-bottomed pot with lid or without, with straight or slanted walls, typically 10"-12": a kitchen staple, like a knife or a pot.
10" frying pan with a lid, relatively high 3" walls and oven-safe handle is a most universal cooking dish. In the very beginning or my independent (from my mom's kitchen utensils, mostly) life I managed to produce a decent clear chicken stock in such pan. Frying in a sauce pan is much trickier. That is because the frying pan, there the cooking goes on the bottom, must have even heat distribution, and in the thin-bottomed boiling pot the heat is distributed via the liquid.
Skillets come in different sizes, shapes, materials and finishes. The shape and the size are matters of use; I prefer larger ones, but browning a handful of almonds in a 12" pan is an overkill. Materials and finishes, on the over hand, are matters of heated discussions, no pun intended. For the even heat distribution a pan must have thick bottom made of a good heat conductor: cast iron, copper or aluminum. Copper is highly reactive and poisonous when oxidized, but provides very even heat distribution. Copper pans are tinned or clad with inert stainless steel. Steel-clad copper pans are great, but tend to be expensive. Aluminum is a great heat conductor, it's lighter and cheaper then copper, and also highly reactive. Aluminum oxide is not poisonous, but alters the taste and look of the food when forms while you cook. Aluminum pans come as all-clad, too. Some are shiny steel, others are matte black hard-anodized (covered with thick layer of inert aluminum oxide) or shiny black nonstick (covered with Teflon). Note that all clad surface means that the copper or aluminum core is sandwiched between layers of stainless steel. The steel can be covered with non-stick layer inside. Traditional cast iron frying pans are heavy, black on all sides, they heat relatively slowly and give off heat evenly and slowly as well. It can be an advantage for some recipes and, especially, with an unreliable heat source, like a campfire. Cast iron pans, if properly cared for, or "seasoned", are non-stick and take very little fat to cook on. I have one but don't like it very much: it takes special process to clean it, it wouldn't cool fast if I need it -- and holding the heat is not an issue with any modern gas stove, its hot handle is not safe around children, or myself for that matter. I use it only for blackening required by some ethnic recipes. I don't care about non-stickiness, either. Nonstick Teflon surface wears off, one should take care never to touch it with metal utensils, nonstick frypans often come with plastic handles, and Teflon decomposes in high heat anyway, with poisonous products. My favorite frying pan is a Calphalon all clad, with an aluminum core and a shiny stainless steel surface, and two short steel handles. It was about $70 with a lid circa 2000 on Amazon. It is 12", so one long handle would be unpractical, and this short handles stay cool enough on the stove top. I use pot holders to take the pan from the stove, but don't get burns if I need to shift it. Aluminum core heats and cools fast, giving me extra control. The pan is dishwasher safe, but gets dull and discolors in the dishwasher. To clean it, I wait until it cools down and fill it with soapy water. After that, I wash it and dry it as any dishes. Some mild abrasives are safe on it and Bar Keeper's Friend brand is recommended by manufacturer. Clean stainless steel is not officially non-stick surface, however, if you first heat the skillet until the rim is hot but not burning to touch, add very little fat and let it heat (or melt), you will be out of trouble. And if something sticks to it, you will not ruin your cookware by cleaning it off.
Griddle is a stove-top pan with low walls, often it's rectangular meant to fit over two burners. I'm shopping for one now, mostly for pancakes.
See also: kitchen, kitchen knives, knife sharpening, stand mixer, coffee grinder.