"The man has many pieces in the green jar."
Translation:Bărbatul are multe piese în borcanul verde.
"borcan" (=jar) is the noun, as found in the dictionnary. You use the article when you refer to a SPECIFIC one (borcanUL = THE jar).
When you have several jars of different colors but only one is green of course this is SPECIFIC so you have to use "borcanUL verde" = "THE green jar"
If you have only ONE jar around Romanians will not consider this specific, as you do not have to explain which one. They would say "Am monezi în borcan". In English you still have to use the article "I have coins in THE jar" (or even "I have coins in MY jar" depending on the context).
This is somehow similar to the english usage of "in life" : We say "the nice things in life" instead of "the nice things in THE life" (as opposite to "the nice coins in THE jar) :-)