In modern technology terms a modem is what is provided by your ISP (Internet Service Provider such as Time Warner, Spectrum, Century Link, etc..) to go from their network to your home/office network. Think of the days when you had to us a modem to "dial" on to the internet with AOL or Earthlink. The modem was built in to your computer but now since everyone has high speed internet with a home network you have a modem from your ISP. The reason the ISP has to "manage" your modem is because the speed of your internet is determined by how the ISP sets up your modem so they like to control that hardware. In our area a modem could connect to the Internet Service Provider's network via a DSL connection (via your phone line), coax cable (like your tv could), or fiber optics. If you connected right from your modem to your desktop or laptop your computer would be directly connected to the internet. Not a good thing since they anyone can access your computer.
A router or wireless router are essentially the same thing. The wireless version (pretty much everyone has this version) provides wireless internet in addition to everything else the router does. A router takes the signal the modem allows the internet to be shared by all of your devices such as laptops, printers, smartphones, and tvs. A router does exactly like its name implies... it routes the traffic from one network (your home) to another network (your ISP). That is why when you set up a printer on your home network it stays separate from everyone else's network. All routers have two basic features to them, a firewall to prevent any outside traffic from getting in your home and a service that assigns each device on the network a unique IP address.
A switch is a piece of hardware that allows one connection to be split in to multiple connections. Really that is all they do but they do it so fast and effective most of the time if you are having slow internet speeds it is probably either your modem or your computer or device. Most people do not have a dedicated switch but an office or business probably does since they may need 10 - 48 computers plugged in.
You might have a modem/router/wireless/switch all in one unit that everyone just refers to as a modem since it is connected directly to your ISP. Most all of the "modems" provided to your by your ISP are these types of units. Some people may have a modem then a separate wireless router, of if you are a business you may have a modem, router, switch and then separate wireless access point. It all depends on what kind of network is needed, the speeds required, and how many users there are.
So, that all being said you can look at your router or modem and know a little more about it and their specific jobs.
For your home networking needs you can look here and find the best wireless router to use: https://www.reviews.com/best-wireless-router. They have some great information on different wireless routers that help you determine the best one for you needs.
If you have any questions please feel free to reach out. We do all kinds of big or small network design, implementation, and troubleshooting.