There are a ways to offer event registration on your website:
The low-tech, free way and manual method is to create a Custom Form on the website using our built in form tool. This will allow visitors to simply fill out the online form and submit it. This then emails you with the registration information and you process manually and followup.
Calendar + EventBrite
You can create events in your calendar, and then enter a URL on the event for registration. This could go to a third party ticketing system (e.g. EventBrite) which will then handle the payment and registration.
You can use third party form builders like Wufoo to create forms (and then embed on any page of your site by copy/pasting the embed code they provide). Wufoo has a lot of guides about how to use their forms for Event Registration.