Here's how it works, at least in the US, Japan, UK, Canada, and Australia. Not sure about other countries.
Typically, you have two types of high speed internet: DSL and Cable. Australia also has Telstra, which uses some type of wierd PPPoA, but I believe they got broken up.
Cable internet typically uses a DHCP connection with MAC authentication. Cable modems are usually manufactured by RCA, Motorola, Scientific Atlanta, Arris, and a sprinkling of other companies.
You are assigned one IP address (the WAN IP) by your cable company, and the modem passes that through to a computer that is connected to it. The modem will also remember the MAC (media access controller) address of the first device plugged into it, and only let that device online (not always true, but usually). This cached MAC can usually be dropped by removing power from the modem for at least 10 minutes, or in the case of Arris (exclusively VoIP modems), pressing the reset button on the computer.
DSL connections use a protocol called PPPoE (Point to Point Protocol over Ethernet). Essentially, you send a username and password to the ISP, and they assign you an IP address. DSL modems are almost always modem/router combinations, as opposed to Cable modems, which are just modems, and have no routing functionality.
If a DSL modem is doing PPPoE, then the modem itself is sending your username and password to the ISP, then taking the WAN IP address for itself, and giving out private addresses to each device connected.
If DSL modem is in what is called bridge mode, then the computer (or router, or other connected device) is sending username and password information, and receiving the WAN IP address for itself.
You can connect a typical home router to either type of connection. The purpose of this router is to split the single WAN IP address into as many private addresses as are necesary, allowing multiple computers to connect to a single internet connection. A hub or a switch will not work for this, as it cannot split the connection, it just adds more ports.