If someone open one ticket, I can send an automatic email saying that the issue was opened?

[Fernando Reimberg]

The client open one ticket for me, than I would like to know if is posible to JIRA send an auto reply email to the person , saying that the ticket was opened with success.

[Sudarshan]

Hello @Fernando Reimberg 
Yes you can, contact your Jira Admin can create a Automation rule for the requirement. something like the below image: (More detail check here)

[Fernando Reimberg]

thank you , I wil check here.

the reporter is the person who opened the ticket, even if the person is not registered into jira, right?

for example, the person is opened the ticket via one project email.

[Sudarshan]

ahh it depends on the configurations.
Have a check with your Jira Admin 

Get a quick reading --> https://support.atlassian.com/user-management/docs/specify-how-users-get-site-access/