I it my understanding that all of these mean a building, as in “man-made structure.” However, I have been told that сооружение and постройка are rather technical. What do they really mean? How are they used?
I'm afraid explaining difference between them is almost a lost cause because their usage is dictated by context. So it would be more fruitful to study instances of their use within context with the help of Russian national corpus for example or just with Google.
Since their translation into English will also be heavily influenced by context, listing their English equivalents has only so much use.
But in order for my reply to not sound completely negative, here're their rough English parallels and explanations
здание - building
постройка - structure
сооружение - construction
здание - normally a solid building in which architectural style can be discerned with amenities for stay of humans; постройка - a light and/or temporary building or structure, often without amenities; сооружение - complex and bulky building or construction for special purposes, often industrial.
There's yet 4th word in this set which is строение. It for all intents and purposes is synonymous to постройка but may also denote a more substantial structure.
But again, their meanings may overlap due to contextual exigencies.
I could list their Ukrainian versions, which may or may not have any correspondence to similar Polish nouns.