DIDs (aka DDIs) – numbers
DID stands for Direct Inward Dialing, while DDI means Direct Dial In, and both acronyms refer to the same thing: A phone number that will lead incoming calls to a device. In our case, a call to that number will be ringing a SIP device.
Normally a customer will port his/her pre-existing PSTN number(s) to his/her ITSP (that is, the customer's number will not make the landline ring anymore, but will ring the end customer's SIP device passing through the ITSP SIP network). ITSPs often have a specific branch of their customer service assisting in the number porting procedures.
DIDs are sought by customers for many reasons: As a primary way to get incoming phone calls (for example, the main phone number of a person or a company, if they have no previous number, or don't want to port it), or as a means to be present in local, regional, or international markets, so as to allow the public to reach a company for the cost of a local call, or to be compliant with regulations that require a company to have a local phone number for customer support.
Also, for each country there are special kinds of numbers with special billing: They can be called for free (for example, toll-free numbers, "800" numbers) from national fixed lines or from both national fixed and national mobile lines, or conversely they can cost a premium fee to be called, a premium that goes in part to the assignee of the number (for example, for pay numbers, hot chatlines, special support lines, "900" numbers, and so on).
In a way, DIDs are the opposite of SIP trunking. A DID provider gets phone numbers assigned or reserved from the competent authorities, and routes inbound calls from those numbers to the SIP devices of its customers.
DID providers can have an agreement with Telco companies to have the numbers routed directly to their SIP servers. Alternatively, they can have special devices, physical gateways, that accept from one side, telephone lines (one at time, or more often T1 or E1 "trunks" composed of 30 voice channels multiplexed in one cable) or cellular network "SIM interfaces", and transform the incoming PSTN or PMLN calls into SIP calls which are then routed to the customer's SIP device.
Each DID is provided with a "capacity" measured in "channels", that is, how many concurrent calls can be incoming on that number (and routed to the customer) before the caller hears the busy signal. Capacity can be from one single voice channel to hundreds.
DIDs have a worldwide market, and multiple local phone numbers from any number of countries and regions can bring incoming calls to the same SIP device, SIP call center, or SIP PBX.
As for routes, DIDs in the same country or region are often offered by a multitude of operators, from first tier big Telcos, to illegal and shady groups, to inexperienced and temporary new companies, with very different inherent parameters of stability, continuity, reliability, audio quality, and so on.
Each ITSP can provide DIDs, at least in a state, country or region, and there are many different global DID providers able to offer regular, toll free and premium numbers from a multitude of countries. Those global providers often buy DIDs from smaller local providers and then resell to ITSPs (that sell to end customers).