When I try googling кружка, I don't get anything that resembles a cup, but it's the first choice when I use Google Translate.
Theoretically, чашка would be an almost complete analogue to the English cup, while кружка is closer to a mug.
Indeed the origin of the word чаша (of which чашка is a variation/descendant) is usually traced to Proto-Slavic čаšа (according to Max Vasmer's etymological dictionary), which is related to Old Prussian kiosi, meaning a goblet or chalice: a cup. Even in modern Russian чаша may be used to poetically refer to that kind of a vessel.
The similarity of cup and чашка can also be seen in other terms. For example, the bra cups are usually referred to as чашечки (a diminutive of чашка) in Russian terminology.
There are a couple of irregularities, though.
For one, in the geometrical sense, чаша is used to refer to any bowl-shaped object. For instance, when referring to a crater, English is more likely to classify it as a bowl-shaped depression, where is Russian term would be чашеобразный (in fact, the insides of the crater are even scientifically called чаша кратера).
Another one is a bit more practical. As @Curiosity mentioned, when talking specifically about disposable plastic/paper cups, the ones shaped like a handle-free glass, Russian uses the word стакан (glass). The ones shaped like a cup with a handle would still be called a cup, however.
On the other hand, кружка is pretty much exclusively used for mug, or tankard, or even stein.
With that in mind, when you are talking about modern-day drinking vessels, you can safely use чашка/чашечка in place of a cup and кружка/кружечка in place of a mug.
That being said, pottery/woodwork/metalwork is wild, and there are obviously certain edge-cases where the difference between cup/чашка and mug/кружка is uncertain at best, and I would hesitate to guarantee that both languages and associated cultures would classify every object the same way.
The difference between чашка and кружка, as a kind of vessel for a liquid to drink or pour from it, is rather subtle and is better expressed in pictures:
This is чашка but never кружка. So, if a cup has apparently narrower bottom than top, and provided with a saucer, it canʼt be кружка.
This is кружка but never чашка. We donʼt use чашка for a cylindrical metal vessel (typical tourist equipment).
This is both чашка and кружка. It is a faience or porcelain cylindrical vessel with a handle. What word is used, it may be even regional (according to my observings, кружка is more used for it in northern regions, but чашка - in southern; but this is definitely not strict).
Also, both words are used for vacuum flask ("thermos") cap provided for drinking from it.
Also, one should take into account that for some vessels, стакан is preferred, mainly for high, narrow vessel without a handle and which form is close to cylinder, maybe with slight widening to top; it is typically of glass or plastic.
There are also multiple cases to use чашка in technics, as a kind of round detail. This use is to be specified separately for each kind of detail.