Сковорода is simply a pan or a frying pan, but not a saucepan. "Pot" can be "кастрюля" as in "pots and pans", but if it is made of clay or ceramics, it will be called горшок, and if it is made of cast iron and is shaped like a pear put upside down, it will be called чугунок (from чугун = cast iron). The most common translation of кастрюля is "saucepan". The word кастрюля was originally derived from "casserole", but, unlike the latter, it refers to a vessel.
I'm going to disagree here. Кастрюля is pot, сковорода is pan, saucepan is сотейник. Saucepan -- is a type of a pan with higher walls for sautéing. Google images seems to agree.
(Native Russian speaker from Moscow originally, living in the US for 22 years now.)
Edit: сотейник, of course.
Firstly, сотейник spells with an 'o' after 'c'. Secondly, as a native Russian who has lived in Russia all his life I can assure you that this word is rarely used these days. More often people call it глубокая сковорода or кастрюля с длинной ручкой, depending on how deep it is. A saucepan is usually called кастрюля when it's depth is bigger than its radius. Кастрюля does not necessarily have a long handle. It may have two symmetrical small handles instead. I guess, in that case, 'pot' would be a more appropriate word for the utensil. Кастрюля made of glass and designed for cooking in the oven is called 'casserole'.
It's what you put saucepans on when you want to cook. Kind of the same as a stove, I'd say. http://www.oxforddictionaries.com/definition/english/hob
I'd say it's widely used throughout Britain. I didn't realise it was not widely used in the US. Found this link about the word if you're curious http://forum.wordreference.com/threads/cooker-vs-cooktop-vs-hob-vs-stove.1422990/
Hob = hotplate. Пилта is the whole cooker/stove. There is a semantic difference although the meaning is understood. Interestingly, Yandex Translate gives пилта as an alternative translation for "hotplate", but I don't think it's fair to say a hob and a cooker/stove are the same things for the purpose of this exercise.
I guess, it is OK to add that sometimes the names of the kitchen items pretty much depend on the place (in the USA, for example) where they are used. http://www.goodtoknow.co.uk/recipes/pictures/27182/150-family-dinners-under-500-calories
In Ireland we use cooker as the generic term, especially for modern appliances. Stove is very old fashioned. Oven is the enclosed part of a cooker used for roasting or baking. Range is typically a unit that provides heating as well as cooking, found very often in farm houses, but there are also modern up market versions in large urban kitchens. Hob is what we call a horizontal cooking area fixed on top of a different kitchen unit like a dish washer. The individual cooking points in a hob are called rings or elements.
Check images for casserole on Google. The word can refer to several type of vessels including classical кастрюля. It can also be форма для выпекания лазаньи or форма для тушения мяса с овощами. Пироги in Russia are seldom (if ever) baked in casseroles. For making a pie a metal sheet (противень - the word derived from the German Brotpfanne) is used instead.
I did: eyeballing it, out of the 100 or so, three or four were кастрюли -- not very convincing. The rest were the almost flat rectangular baking vessels from glass, ceramics and metals that go into the oven to make casseroles and lasagnas. With handles and without.
Форма для тушения/выпекания -- sure, that's exactly it. I mostly agree about противень (baking sheet) and pies. Some exceptions though would be the cases where you need to make something that takes liquid or easily flowing dough, such as бисквит, шарлотка and so on: you would use a casserole for those. I actually make a cabbage pie in a medium sized casserole rather than on a baking sheet, it's just easier to handle.
I appreciate the etymological details you frequently include in your posts, by the way. Not being a linguist, I find them very interesting and some are totally unexpected.
We tend not to have pots in the UK these days, nor stoves (apart from rich folks who inhabit country houses and expensive parts of London, that's just the stoves part, and they tend not to use the word stove, but the brand name 'Aga') Pots went out years ago as a word for a cooking utensil, saucepan is generally used for anything that goes on the top of a cooker (usually known as the hob) and casserole dish or roasting tin for anything that goes in the oven part of a cooker. So the term put the pot on the stove would probably only be used by someone jokingly or over the age of 80, as would an old fashioned idiom 'put wood in th' hole' (probably only in the north of England) for close the door.