I'm new to anime and my friends really like it. They really like to say to me "notice me, senpai!" and I don't get the reference. I could be walking to class and one of my friends would say "Can you go to the canteen?" and I would say no I can't and then, they will say "NOTICE ME, SENPAI!" Can anyone shed some light on the "notice me, senpai" reference, please?
This is a meme in the English-speaking internet anime community. This particular phrasing appears to have come into existence around December 2013 (according to Google Trends), though variants are known to date back to 2012.
It does not directly reference any particular work of anime or manga. Rather, it is inspired by a reasonably common trope of anime/manga writing about school-age children, whereby a character (usually a girl) will have a personal or romantic interest in an older student at their same school (which is what a "senpai" is in this context), but instead of doing a dag-blasted thing about it, just silently hopes that the senpai will pay attention to them. This narrative trope is particularly common in shoujo media.
Goodness knows why your friends are exuding memes in real life; you would do better to ask them about that. As a meme originalist, I cannot help but notice that the contexts in which your friends are reportedly using this meme don't make much sense.
If you know what senpai means you may understand. Senpai means upperclassman or someone you look up to. This phrase is quite common in anime which have a school background. It is generally used when someone wants another student (who is older than them) to notice them or acknowledge their presence. I once found this online:
It's a Japan thing. Senpai means someone you're suppose to look up to. Maybe a fellow student or colleague. Maybe your teacher. All in all 'Notice Me Senpai' means you want someone older than you to like you.
An example: Say a girl has a crush on a boy who is one class ahead of her at school. To the girl the guy is a 'Senpai'. Now, to impress him, she tries to act cool in front of him, cracks jokes, does some wierd stuff. Now say the boy is walking towards her. So she silently wishes "Notice me Senpai !" But the boy doesn't even look at her twice. So she would say to herself "Senpai why don't you notice me?"
But many times in real life, people say this to each other just for fun, maybe as in the case of your friends. Since you are new to the anime world, this may seem a bit odd but a few anime later, you will understand when to use the Notice me senpai reference yourself.
Although when your friend would say that when you refuse to do him/her a favour could mean that if you would 'notice' how pretty/awesome your friend is, you would certainly want to help him/her, and get on their better side... just an idea. You might want to ask your friend itself...
A much shorter answer: in a lot of manga and anime there is a character that likes a senpai, an upper classmen, however, no matter what they do, they can't get noticed by the senpai. So, the line expresses their desire to be noticed by their senpai. It has become a line to make fun of people when they are doing something for attention or interacting with their crush.