I’m using
Configure SMTP plugin for wordpress, when I used it in a local server (localhost) it works properly and send me the test email which confirms that it is working. But when I used it in a hosting site (I used Hostgator) it gives me an STMP error. 
What is the possible cause of this error? and what is the possible solution?
Thank you.
Based on the second error, and the words “verification failed”, and “sender verify” failed, my guess would be that you have not set up SMTP authentication correctly. You are relaying your mail through Hostgator, so they may have specific hostnames / ports that you need to authenticate correctly with to get your mail to flow. I’d suggest syncing with Hostgator on this.