And you should. You own your own domain name and are responsible for renewing it. In case you are wondering, a domain name is a unique name that identifies an internet resource such as a website. For example, this is the address you type into your address bar in your browser to get to your website (www.domainname.com), or the email@example.com that you send email to. Beyond Indigo simply hosts your website, mail, or other services associated with your registered domain name. If you leave BI as a client, you still own your own domain name and can easily transfer it to another hosting provider using any Registrar service.
However, as stated above. You are responsible for your own domain name. Usually, when you register your domain name, you can prepay for one year, 3 years, 5 years, 10 years, depending on your provider (i.e. GoDaddy, Network Solutions, Register.com, etc.). When the renewal time comes, you will get a notice from them to renew your domain name. It's important that you renew within the time period they specify, or else your domain name will expire and this will cause your services to faile (i.e. your website will no longer function and neither will any other service associated with that domain name). It's like paying your electric bill. Do it on time, and your lights will remain on.
Please make absolutely sure that the notices you receive are from the original place of registration. There are many phishing scams out there that ask you to renew with the sender of the notice because they are banking on the fact that you've forgotten who you've registered with. Don't forget. Make sure you know. And if you don't know, ask Eric or Noelle, they can do a simple lookup and find out for you.
If you want to check for yourself, enter the name of your domain here. The name of the registrar of your domain will be on the second line.