Step 1: What Each of Your Devices Do
A router (wireless or wired)
A network switch
Many times when you have a wireless router supplied by your ISP (internet service provider), it has the modem built in. You can see two routers and a switch in the pictures.
Here is the basics of what each device does:
The modem is the device that sends all of your data to your ISP so that you can communicate on the internet.
The router "routes" between different networks. In the home setup, there will generally be the outside network and your home network (inside network). In addition to routing traffic between two networks, your router will generally have NAT (network address translation) and a firewall. NAT is used to save public IP addresses (all of your home IP addresses are condensed into one public IP address supplied by your ISP). It also provides some level of security. Your firewall decides what traffic can get into your home network.
The switch allows you to expand your wired network by giving you more ethernet ports that you can use. It switches which ports are communicating at one time at a very high speed.
I will not get into how each of these work, as it took me about 1.5 years of very intensive learning to understand networks in the way I do today.
Step 2: IP Addresses
Most routers will automatically assign each computer an IP address through a process called DHCP (Dynamic Host Configuration Protocol). When your computer joins the network, it basically sends out a message to every computer on the network that says "can anyone tell me what IP address I should use?", and the router will reply with the addressing information.
To find out what IP address is assigned to your computer, go to command prompt, type "ipconfig", and press enter. This will give you a display of all the settings for each network adapter on your computer.
It is also worth mentioning that this is examples of IPv4 addresses. Some large companies are now switching over to IPv6, which is completely different from IPv4, but that shouldn't affect your home network. (At least not any time soon.)
Step 3: DNS Servers
Step 4: Conclusion
You can see a topology in the picture.
If you have any questions, feel free to comment below. Please note that this is not meant to be an in-depth exploration of networking, it is just a guide to give people a basic understanding of how their network works in order to help them troubleshoot.