Definition: HDMI stands for High Definition Multimedia Interface.
HDMI can pass video resolutions from 480i up to 1080p. HDMI can be implemented on Televisions, AV Receivers, DVD Players, Blu-ray Disc Players, HD-DVD Players, HD Cable Boxes, and HD Satellite Boxes.
HDMI also includes provisions for HDCP (High Definition Copy Protection). This allows content providers to prevent their programming from being illegally copied.
HDMI can be adapted to DVI (Digital Video Interface), via adapter cable or connector. However, the device that has the DVI connection must be HDCP enabled for the signal transfer to work.
There are several versions of HDMI that have been developed over the years. Depending on when you purchased an HDMI-enabled component (HDTV, DVD player, Blu-ray Disc player, etc...) would determine what HDMI version you have. All newer versions are backwards compatible. You can still use newer versions of HDMI with components equipped with older versions, you just won't be able to access the all the content features of the newer version(s).
HDMI 1.3 - This version includes improvements in both audio and video capability. With the advent of Blu-ray Disc and HD-DVD, version 1.3 adds the ability to transfer the digital bitstreams for the new high resolution audio formats: Dolby Digital Plus, Dolby TrueHD, and DTS-HD.
HDMI 1.3a - In addition to the above audio improvements, HDMI 1.3 and 1.3a increase the amount of video bandwidth that can be transferred from a source to a display. This means, that in addition to the standard 24-bit color depth we are used to, HDMI 1.3 and 1.3a have the ability to transfer color depths up to 48-bits, and can accommodate resolutions much higher than the 1080p resolution standard that is in use today.  For additional details about HDMI check out: HDMI Facts - An Overview of HDMI
NOTE: As of January 1st 2012 references to HDMI ver numbers on new products are to be eliminated - Read Official Announcement.
For the purpose of communicating HDMI cable features to consumers that are now five cable categories: HDMI Standard, HDMI Standard with Ethernet, HDMI Standard Automotive, HDMI High Speed, and HDMI High Speed with Ethernet. For details on each category, refer to the Official "Finding The Right Cable" Page at HDMI.org.