VoIP is a communication language. It is the general term for the transfer of voice and, more recently, video over an IP Network.
All carriers, including AT&T, Verizon, Comcast, and others, use VoIP to interconnect calls globally. So when you make a traditional phone call today on a traditional phone system, it is in fact a VoIP call. Even a call to or from your cellphone is VoIP!
ATA - The simplest and most common way is through the use of a device called an ATA (analog telephone adaptor).
IP Phones - IP phones connect directly to your router and have all the hardware and software necessary right onboard to handle the IP call. Wi-Fi phones allow subscribing callers to make VoIP calls from any Wi-Fi hot spot.
Computer to Computer - This is certainly the easiest way to use VoIP. All you need is the software, a microphone, speakers, a sound card and an Internet connection, preferably a fast one like you would get through a cable or DSL modem.
In basic terms a Hosted Private Branch Exchange (Hosted PBX) is a telephone exchange system that is built, delivered and managed by a specialist provider using IP telephony - so it is provisioned and delivered via the internet instead of via traditional PBX equipment that would once have needed to be installed in-house.
A new, small, under-capitalized business can benefit hugely from hosted/cloud systems due to the reduced up front cost of technology.
There are some additional, real advantages should you shift your communications technology to the cloud. Companies with multiple locations, remote offices and teleworkers can hugely benefit from Hosted PBX.
Not only are users stuck with a significant upfront cost, but they're subject to hefty ongoing costs too. Traditional PBXs use the old-fashioned (PSTN / ISDN) landline network. This means that every single user requires a separate phone line with a seperate phone number, compounding monthly line rental costs.
Furthermore, the legacy phone networks can often have high call rates and flagfall charges. If you use the phone alot, this will quickly compound your monthly charges.