Connecting a home computer to the internet only requires a modem, which is a device that allows a home computer to connect to the internet via a specific type of data transmission system. Before high speed (broadband) internet, modems connected to the internet via [dial-up] modem (MOdulator - DEModulator) that transmitted digital data at the computer to acoustic information via the telephone lines.
The telephone modem has long fallen by the wayside for most people. With the ubiquity of broadband internet in homes, having a home network is a natural progression that only requires the addition of a router. A router is a device that combines a switch (a device that allows devices to communicate with each other) with a gateway (a device that allows one network to connect with another network, like the internet) and is installed downstream of the modem. The router allows multiple devices (personal computers, mobile phones, etc) to use a single internet connection simultaneously.
An important advantage of having a router between internet and your computers is the router's hardware firewall capability. A hardware firewall shields your private home network from external cyberthreats. See:
- PC & Account Security - Part 1 - Hardware Firewalls & Network Encryption - Runescape Weekly
- Hardware Firewall vs Software Firewall | Network Security | TechTerms
Residential Gateways are devices (typically provided by an Internet Service Provider - ISP) that combine multiple functions including modem, switch, and wireless access. Depending upon the ISP, some features built into the residential gateway require additional monthly subscription fees to unlock. It may be cheaper to provide your own devices to get around the ISP's money grab.
A home network is considered to be an intranet, meaning that the devices can communicate with each other and are peer-to-peer in that there is no hierarchy (ie, each has the same importance/priority). Most home networks have both wired devices and wireless devices and many of those devices require the ability to connect to the internet. While the standard for wired connections is Ethernet, there are a variety of wireless standards that have evolved over time. Ethernet is the generally the fastest and most reliable means of data communication although some forms of wireless communication can be faster.