There are several websites that let you sing karaoke online. While most allow you to join for free the free memberships allow access to a very limited number of songs. For a fee (usually monthly) they will give you access to their entire library of thousands of karaoke songs.
Along with allowing to to record your audio performance most online karaoke sites let you record a video of yourself using a webcam. Your performances can be shared with others and sometimes there are rating systems in place where others can rate your performance and post comments. There may even be singing contests to join.
Getting set up at one of these sites is usually pretty straight forward with easy to follow instructions. The script or applet used is designed to detect your microphone and webcam although there may be some system settings you need to change.
Rarely will there be any special equipment required. Virtually any microphone that will plug into your computer can be used. The same applies to the webcam. This isn't to say that you can't upgrade your setup. Some people take their singing seriously and opt for high end equipment.
It's not difficult to upgrade to a better microphone. A good USB condenser microphone can make a huge difference in the sound quality of your recordings. These mics can be purchased online or at most electronic stores and typically install themselves when you plug them into a USB port.
Some people will take it a step further and add a small USB mixing board. These allow you to use any type of microphone and fine tune the microphone input. Some mixers allowing you to add effects like echo, reverb, etc.. These mixers are also very easy to install but may take some practice to use correctly.