Two-factor verification is a service that adds additional layers of security. Normally you enter a username and password in order to access a website or application. With two-factor you would be prompted to enter one additional authentication method. With multi-factor authentication, you would be prompted to enter more than two additional verification methods. There are many different methods of two-factor authentication from Microsoft, Apple, Google, and others.
Here’s a Google video about two-step verification basics, which provides a good idea of what’s involved.