There are a variety of reason you may want to re-brand your company, both online and offline. For the purposes of this article we are going to focus on online re-branding.
Typically re-branding means changing the company name and logo. These things can be easily replaced on your existing site and social media profiles. However, you will also want to change your domain name as well. Static HTML sites can be easily transferred, but database dependent sites can take more work. WordPress, the most popular CMS, can be easily moved to a new domain by follow these directions. Other CMS solutions will require a different set of instructions, which are usually provided in the FAQ section for the CMS.
Beyond just buying a new domain and moving your content, you will also likely want to set up your old site to redirect automatically to your new one. Typically a 301 redirect is recommended (assuming this is a permanent change).
1. Change name and logo in site content.
2. Transfer site content.
3. Setup 301 redirect from old site to new.