Wired or wireless, what are you going to pick? Today most printers have both options builtin and sometimes you can use more than one connection for redundancy.
If you go with wired, your connections can be usb or ethernet(via router). You can use both types also.
A wireless router will be required if you want to utilize your wireless printer. If you want to use- USB then that is an option too for those with desktops that have no wireless builtin. Also, you can use the Ethernet port if your printer supports it to make the printer available over the network just like wireless.
HP E-Print and AirPrint.
“ePrint – Use this HP feature to print from anywhere with a Web connection by sending an email to your printer’s email address.”
“With AirPrint, you can print from an iPad, iPhone, or iPod Touch to HP ePrint-enabled printers via Wi-Fi.”
-If any questions come up feel free to comment on the topic.