You can download the patchs from the links below or as an automatic update through Battle.net.
Changes from v1.07 to 1.08:
Adds support for Battle.net gateways.
Changes default Battle.net font so that it is easier to distinguish certain characters while in chat.
Adds command completion to Battle.net chat. It can be accessed by pressing the "tab" key.
Fixes some minor battle.net issues.
Displays cancel dialog in situations when server is busy.
[PC Only]. Multiplayer character files are now located in the Diablo directory. These files are visible, more descriptively named, and may be moved from one computer to another.
Color cycling On/Off now appears in options menu. This will help some low end systems and systems with troublesome drivers.
Spawn is now updated.
[Mac Only] Improved Mac vs. PC multiplayer, modem, and direct connect.
[Mac Only] Added Input Sprocket/multibutton mouse support.
[Mac Only] Added option to share processor time.
[PC Only]. The registry key HKEY_CURRENT_USER is used rather than HKEY_LOCAL_MACHINE.