"I am a boy."
Generally, "chłopiec" is used for younger boys, and "chłopak" starts about the age of a teenager. There is no clear border though. Same goes for "dziewczynka" and "dziewczyna".
"chłopak" and "dziewczyna" can also mean "boyfriend"/"girlfriend", although I wouldn't interpret "Jestem chłopakiem" on its own as "I am a boyfriend" ;)
Cases. That's a very wide topic, so generally I'd suggest looking for topics about cases here: https://www.duolingo.com/comment/16296174
As these two specific words are concerned, "chłopiec" is Nominative (the basic, dictionary form) and "chłopcem" is Instrumental. So "chłopiec" will mostly be used when it's a subject of the sentence, or in sentences like "This is a boy" (To jest chłopiec). Instrumental will mostly be used in sentences like "I am/You are/He is a boy". A quite detailed guide to those constructions is here: https://www.duolingo.com/comment/16373167 - you can just start with Part 1.
This is what I found for an explanation. I hope it helps others. I'm still very unclear about the meaning. "In these initial lessons you may sometimes stumble upon nouns in the Instrumental or Accusative. For now, you should be able to form basic sentences with the help of the hints attached to particular words. In the following skills we will gradually introduce you to the rules governing Polish declension."
Well, Wiktionary usually goes into this subject; for example, https://en.wiktionary.org/wiki/by%C4%87 explains it for "być" (the verb in question here)
One of our esteemed moderators has written quite a lot about this subject here: https://forum.duolingo.com/comment/16373167/A-guide-on-X-is-Y-and-This-is-Y-constructions
A suggestion for a complementary/alternative terminology to cement the concepts:
For native English speakers the concept of'direct object' just as'indirect object' (or simply 'object') might be more meaningful and memorable than 'instrumental', even if they mean the same thing.
In English nouns are not inflected for grammatical case, as is the 'case' in Polish.
Except that various verbs take different cases for their indirect objects; some take Instrumental, those which take Accusative take Genitive in the negative, some always take Genitive, some take Dative.
It is still necessary to determine which "indirect object" is being used, in which case we might as well use the formal case name system.
Is "a boy" in this sentence a direct object? I was under the impression that almost always the direct object takes Accusative in Polish (and sometimes Genitive), and then the indirect object takes Dative, but I never thought of such a sentence as here as having an object. But syntax was never my favourite subject.
No, "chłopcy" is just plural for "chłopiec".
The plural form of "chłopak" can be "chłopacy" (but that's rather rare) or "chłopaki" (the common one). However, the word "chłopaki" is a weird grammatical exception, because for reasons unknown to me it is treated like a 'not masculine-personal' noun. So you have "ci chłopcy" (these boys), "te dziewczynki" (these girls)... and somehow also "te chłopaki" for "these (older) boys".
Ah, so suddenly they ARE instrumental. (I just asked a question about the sentences in the last set which weren't). I recognise this now because I'm coming back from having been a bit further on, but it hasn't been explained, and it seems very strange that this one has to be instrumental when 'She is a woman' and 'I am a man' in the last set weren't so presented.
Yes. This is instrumental case, often used after the verb "być" ("to be").
For further reading, please take a look at https://forum.duolingo.com/comment/16373167/A-guide-on-X-is-Y-and-This-is-Y-constructions