Hosted pbx is a service several companies (called ITSP, Internet Telephony Service Providers) offer to give any individual or business a professional telephony presence using the Internet. Software running on the ITSP network offers to deal with voice and fax calls (and sometimes SMS) in a way similar to what a mail server does for your email.
You can add a normal or a toll-free DID (Direct Inward Dialing) number to a hosted pbx in order to ring multiple phones, accept voicemail messages, faxes or text messages (SMS) from those to whom you provide the phone number. You can also progrtam service with relation to time of day, for example routing calls outside opening hours of your business to voicemail boxes.
Many interesting services are available on a pbx, such as forwarding calls according to time, busy state of a particular phone or decalred availability of an extension. For example at night and during holidays, calls can be automatically routed to a different extension, forwarded to a mobile number or to voicemail.
Pbx, hosted or running on local hardware, are often used a switches, a way to route calls.
IVR, (Interactive Voice Response) can be used to route calls or collect authentication information. On a local pbx, you can require the caller to enter digits or route the calls according to the caller id information they are sending.

