Wired: In simple terms; the communication between two devices via cables.
Wireless: the communication between two devices without cables.

Now, is it that simple? Not exactly, each method of networking has its own pros and cons.

Wireless networks do not use any form of cable. The transmission of data (your files, music, printing to the printer etc) occurs over radio waves… just like cordless phones or the Bluetooth headset that came with that snazzy phone you purchased recently. The major advantage of having a wireless device is the mobility and freedom that comes with it. Moreover, there is less clutter and fewer wires to worry about. But, you sacrifice in most cases on speed and security. The security issue is important and at least some form of basic or weak encryption should be implemented. I strongly recommend you to research on high encryption levels and implement it. This will at least deter the nosy neighbor to leech your internet connection.

Wired networks on the other hand have been around for some time now. Officially known today as the Ethernet, the cables usually connect these devices using CAT5 cables. The speed and security in this scenario are greatly enhanced. The latest Ethernet routers can support up to 1000Mb/s or a gigabit/sec. that's 10 times faster than the widely used 100 Mb/s router. Moreover the over all cost of a wired network is lower, provides high performance and better security than wireless networks.

The choice depends on your day to day activities. Wireless networks won't become mainstream anytime soon in office environments. But as home users, wireless networks have become the choice. A little sacrifice in speed or security as a residential user is minimal but for an enterprise environment – well, it becomes a whole new ball game… and, pretty detailed too.