A Hosted PBX generally involves installing some sort of remotely attachable handsets (usually VoIP based, in today's world, but "Centrex" service does this well in the analog world, too) at a site, then putting the 'brain' of the PBX, along with it's primary PSTN connections, into a different site, usually controlled by a vendor.
There are several pros and several cons involved, but a lot of the answer comes with your level of comfort with the vendor, and your level of cost involvement.
In the end, the golden rule of telephony development is this .... "Do Not Mess With Dial Tone". Users depend on it, every day- and businesses live and breathe by their phones. If you find a reliable, cost-effective solution, then that makes a good fit- any chance of unreliability, and you are risking the business.
In a hosted PBX, there is one extra moving part- the link between your site and the provider- that you *must* ensure. If that should fail, all your phones will be down (unless you have some sort of backup line arrangement and local PBX hardware to fall back on).
Specifically, here are some pro's and con's of hosted PBX's:
1. Generally, a hosted PBX is less expensive to the end user. Most hosted PBX vendors work out some sort of 'pay per minute' or 'pay per handset' plan, and, since they put many different customers on their enterprise-grade PBX at a central site, are able to pass on savings. For a small site, it's very hard to beat a hosted provider's cost model, unless you've got a large number of handsets, or specific application needs that drive up the price.
2. Maintenance is built in. Hosted PBXs today generally use VoIP hardware- so handset Move,Add,Change work is done by the end user with no more difficulty than moving a PC. The wiring at your site is your LAN, and LANs have a generally high reliability factor. Most changes, therefore, are done at the PBX level- and the vendor can again leverage economy of scale- the changes are generally simple and done via web browser, so they can include the 'maintenance' for free, and get some high-level support on every problem.
3. High reliability of trunk lines, and generally lower cost per minute. Again, through economy of scale, the provider can almost always get a better per-minute rate than a small office can negotiate with the local telco, and can easily afford to have redundant call and network/PSTN paths by sharing them with multiple customers. Having a trunk failure from a hosted PBX center would be horrific, affecting potentially thousands of customers- the vendor simply won't let that happen. (or shouldn't).
1. Loss of flexibility. The hosted PBX company makes their money by providing a fixed package of services and devices to it's customers. If you want telephony applications that aren't on the 'menu', the answer may very well be 'no'. Don't like your handsets- you can't change them (beyond a range). Need to change your long distance provider? Forget it- you won't have one to select. In some cases, the vendor will own your number, making it difficult to change vendors without changing your business telephone number.
2. Reliability. As I mentioned above, you're now dependant upon your local LAN and the WAN connection between your site and the vendor. If you've got a small office on a tight budget, failures of either one may take a few hours- or days- to resolve, as you won't have the local resources to apply to them. On top of this, the cheapest WAN method is seen as the Internet- and the Internet itself may not be reliable. Call quality may suffer if your office is doing a lot of Internet traffic, or your Internet provider is having problems.
3. Business reliability. The best rates for hosted PBX companies come from startups. This has it's ups and downs- sometimes, that can be great, providing personal service at a decent price. But, a lot of startups fail- and when they do, your phones go with them. Check your contracts carefully.