Home networking is almost everywhere these days, and most of us have had to learn the basics the hard way. Lost connections, hacked passwords, and interminable calls to customer support can all cause headaches, and while there's no foolproof way to avoid any of them, it's still true that knowledge is power. I'll cover the basics of networking and show you how to change your settings.
You and your router
Your router is a piece of hardware that takes the internet connection provided by your Internet service provider (ISP -- the company you pay every month) and splits it up, or "routes" it to different devices using wires or Wi-Fi. You'll need to access the router in order to tweak your network settings. There are far too many brands and models for me to try to explain them all in detail, but they do share quite a bit in common. Just remember that you may encounter slightly different terminology or tools; if you get lost, check with your router's manual, online help, or -- if you're really stuck -- customer service line.
Presumably, you've already set up your router using the manufacturer's instructions and information provided by your ISP. You'll need to dig up those instructions and information, so if it's not available, check online or contact customer service for a refresh. Just tell them you're setting up your router and they'll tell you what you need to know.
Most modern routers use Web-based administration tools, so fire up your favorite browser and direct it to the address that came with your router. (Two common addresses are 192.168.1.1 and 10.0.0.1, but yours may be different.) You'll need to supply a username and password; again, this should be supplied by the router's manufacturer, but if you can't find it or you changed it and then forgot it, you will need to reset the router to its factory settings, but you should really only do this if your network isn't running properly, as it will require you to run through the setup procedures all over again.
Basic network settings
Once you've accessed your router, you can explore your options. You can access these settings in your router's menu typically by selecting Internet settings, Basic settings, or something similar.
Your IP (short for Internet protocol) address is kind of like a phone number -- it's a unique identifier. Some ISPs provide users with their IP address dynamically, and it can change from time to time as users disconnect and reconnect to the network. Other ISPs provide users with locked-in IP addresses; these should be entered when setting up your router and shouldn't change. The same is true for the subnet mask and gateway IP address, each of which are used to connect to your ISP.
If your IP address is like your phone number, your DNS (domain name server) addresses are like phone books. ISPs usually provide you with two or three to enter as you set up your router, but there are now many other free DNS options -- see how to find out of one of these could work better for you here.
The MAC address is a unique identification number attached to networking hardware, somewhat similar to a serial number. You shouldn't have to change this unless you get instructions from customer service to do so.
There are two layers to home network security: Keeping unauthorized users from accessing your internet connection and protecting the machines connected to your network from outside intrusion.
If your network is all wired, you should have no problem with unauthorized access to your connection (unless someone sneaks in and plants a cable while you're out, but this isn't a common problem). Wireless password protection is key to keeping wireless users from poaching access and potentially getting you in trouble with illegal activity. Your best bet is to use a long string of random characters, as most devices just require one-time entry for network passwords. You should also change it frequently, especially if you live in a dense neighborhood or suspect that a neighbor may be trying to crack your security. Changing it is simple; just select Wireless setup or Wi-Fi settings and change your password from there. If you've never changed it from when you first installed it, do so now.
All open networks are vulnerable to intrusion, and yours is likely no different. Most routers come with built-in firewalls, which are barriers to outside access. You are almost certainly better off leaving your router's firewall settings alone, unless if you hear otherwise from a customer service representative from your ISP.
Connecting computers and other devices
Of course, the router's main purpose is to connect your computers and other devices (like game systems, smartphones, tablets, and so on) to the Internet. Most use DHCP (dynamic host configuration protocol) to assign local IP addresses to the devices on the network, and this works pretty well for most uses.
Sometimes, though, it's better to have one device (like a networked printer or hard drive) use the same local IP address at all times. Different routers use different language to access this feature, but it's not so hard. Look for LAN settings, Local Network Settings, DHCP Settings or browse through your interface until you see DHCP or Address Reservation or something similar. If in doubt, check the manual or customer service before making changes. To add a permanent address to a device, you'll need to find that section in your interface, then select Add device or whatever your router says. This should bring up a list of devices connected to your router, and it may take some effort to figure out which one is the one you want. Computers are usually pretty straightforward, but your best bet is to check the network settings of the device to see what its MAC address or current IP address is, as that should be unique. Select that device, then add a permanent IP address and a nickname to help you remember it in the future.
Those are the basics of home networking. Of course, there's much, much more you can do, but this should help with basic security and connection issues.