Typically, “Hosted” PBX (“hosted solution”) is when you “rent” a PBX space from the service provider. The benefits are obvious - you need to know very little and provider will maintain the system and will have all the headaches associated with it. However, your feature set it typically more limited.
There is also another type of “hosted” PBX - when you install you own PBX in our data center. This is common if you have multiple locations.
Finally, when you own a PBX - you typically have a lot more features and easier way to customize them - i.e. change music-on-hold based on the season, etc.
