First of all, we’re inundated with the word "hybrid" in all sorts of industries. What does “hybrid” mean? Well, it just means a mixture of two things. Reese’s Peanut Butter Cups are "hybrid." Hybrid cars are hybrid, because they use battery-powered motors and mechanical engines. So what's hybrid hosting?
To make things a little more interesting, there are actually two definitions of hybrid hosting that you're going to commonly see in cloud-related articles and white papers. One is the idea that hybrid hosting means adding cloud services to existing on-site legacy systems, and using a mix of in-house physical equipment and remote vendor setups (this resource from TechWench provides a little more on this concept). But another one that's probably dominant today is the idea of hybrid cloud hosting, which is the idea that you're taking part of one cloud hosting service, and part of another: specifically, a mix of public and private cloud solutions.