I have a problem and I want to google it but I don't know the word for it. To be honest I don't know the word in my mother language either.

this is the problem: besides my nails, sometimes a very small piece of nail -like a thin stick - grows. If I forget it and don't trim it using a nail cliper and it get pulled off by accident, It gets very painfull.

Is there anybody who knows a name for this stick-nail or the name of the problem?

I know I didn't give a good explanation but I don't know how else the tell it.
This is called a 'hang nail'
This is called a 'hang-nail'
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 abbie1948's reply was promoted to an answer.

Thanks pals
Honestly. It's a hang-nail!
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From the Oxford medical Dictionary

Hang Nail - A small piece or sliver of skin which hangs loose, near the root of finger nail.