Q:

Plastic part on the tip of a shoelace

What is the common English name for that "plastic part on the tip of a shoelace" which helps to easily insert the lace into the shoelace holes? (I'm not sure that's the correct name).
Edit: There are also metal parts of the kind.

A:

It's called an "aglet". 

a metal or plastic tag or sheath at the end of a lace used for tying, as of a shoelace.

I've known and used this term for years but whether it can be considered "commonly known" is up for debate. It's certainly not unheard of. 

A:

"Aglet" is a good word and one that I didn't know, possibly more correct and specific (thanks Catija) but I have always heard them called a "ferrule". Generally it is a ring or tube reinforcing something prone to splitting or fraying. Sometimes it is heatshrink tubing applied to the end of a rope, sometimes a little aluminium tube squeezed onto the end of a bicycle's gear cable, sometimes the brass ring on the top of a chisel's wooden handle where it is struck by the mallet.

A:

While "aglet" is the technically correct term, it is one that many people do not know (as evidenced by how often this question is asked, even by native English speakers). I would instead simply use the word "tip." If you need to specify, you can even say "shoelace tip." 
I will also note that "ferrule" derives from "ferrous" which means "made of iron metal." Hence the term "ferrule" is more often used for a metal tip. However, it is also an uncommon word, so I would actually just suggest saying "metal tip" or "metal ring," whichever is more appropriate. 

