Blu-ray is a type of disc that High Definition Video & Games can be played from due to the amount of space available on the disks, more than 5 times that of a normal DVD.
The reason they are not compatible with other drives is that they use a blue laser (DVD uses red) to read the data off the disks, hence the name blu-ray....
You would be better off with a TV capable of displaying HD content for Blu-ray movies but it isn't compulsory. The Playstation 3 for example is uses Blu-ray and it can be used on a normal TV via composite or scart.
HDMI stands for High Definition Multimedia Interface. It's a cable & connectors made to a specified standard to carry High Definition signals between compatible devices.
I can't recommend a Blu-ray writer as I don't have one... they're too expensive at the minute. Here's a selection of from click here