Hello Juan!
Thanks for checking out The Events Calendar! I’m happy to help out. In order for a user to sign up for an event, you’ll want to look into using one of our ticketing add-ons. (These will work with just the core Events Calendar plugin, by the way.)
With one of those add-ons, your users will be able to access their account or purchase history page which will list all the ticket purchases they’ve made. Here’s an example of a ticket purchase made with the WooCommerce add-on:
http://d.pr/i/1dJui
Let me know if you have any other questions or if I can help with anything else.
Kind Regards,
Jami