"A cucumber" is a a singular noun; it means a single cucumber fruit. "Give me a cucumber". "Cucumbers" is a plural noun; it means two or more cucumber fruits. "Give me those cucumbers".
"Cucumber" (without an article) is a collective or uncountable noun, like sand. You can say "have some cucumber", "raita contains yoghurt and cuc
KrisBlueNZ, Thank you very much for your explanation! It's very difficult for me to figure out if a noun is countable or uncountable. I always check words on a dictionary, and on the entry for this word "cucumber", it says "[countable/uncountable]. So every time I come across a noun like this, I ask native English speakers here. Maki