The 's indicates possession.
So in the case you stated it would be Kyle's dog is cool because Kyle owns the dog and has possession of it.
There's also another thing you should know about the 's for possession.
If the person's name ends with s (eg. Amoss) then you don't add an extra s but just put the ' at the end of name name.
Eg. Amoss' dog is cool.
The other time that Kyle's or Amoss' can be used is as an abbreviation for Kyle Is or Amoss Is.
I can't quickly think of any time when you'd just put an s at the end.