1. There's no word to explain it. = Correct. There is not one single word to explain it.
2. There's no words to explain it. = This one is tricky.
There's is a contraction for 'There is'. There is no words to explain it.
This is incorrect grammar as it should be: There ARE no words to explain it.
However... and this is where it gets tricky.. you will often hear people substitute There's when grammar calls for There are..
Ex: There's reasons for my madness! There's five people in the bathroom!
Technically, the second sentence is incorrect. I advise you to learn it as "there are no words to explain it", however, you will hear 'there's' misused---- a lot.
There's just some weird English things that you have to accept!
----please note this pertains to American English. The Brits are writhing in pain right now.