By definition a firewall is device or function that permits or denies network access to a device. This can be in 2 forms: hardware or software. Hardware would be a device in your home that is outside your computer and generally hooked to the network in some way, shape, or form. This device could be a stand-alone firewall box most commonly found in business offices, or built into your internet companies router. Software, on the other hand, is something you download or install on a personal computer. This can be in the form of the all-too-common "Microsoft Windows Firewall", Norton Antivirus Firewall, or any stand-alone piece of software.
The importance of a firewall is pretty big. Without a good one "guarding your network", anybody with minor computer knowledge can easily gain access to all your personal files on any computer connected to your network. Most of the time this doesn't involve them doing some voodoo magic with your mouse/programs/operating system to make it seem like they've come alive, no. This usually involves them leaving back doors on your systems to easily find their way back in, if need be. The attacker can also steal sensitive information stored in your internet browser, like saved passwords, names, addresses, etc etc.
This is an image of where a firewall should be located, in some way, in your network to provide the most protection.
All of the routers from internet service providers (like At&t's, shown below) have built in firewalls! So you're most likely secured against basic intrusion attacks right from the get-go.
