What does %~d0 mean in a Windows batch file?

The magic variables %n contains the arguments used to invoke the file: %0 is the path to the bat-file itself, %1 is the first argument after, %2 is the second and so on.

Since the arguments are often file paths, there is some additional syntax to extract parts of the path. ~d is drive, ~p is the path (without drive), ~n is the file name. They can be combined so ~dp is drive+path.

%~dp0 is therefore pretty useful in a bat: it is the folder in which the executing bat file resides.

You can also get other kinds of meta info about the file: ~t is the timestamp, ~z is the size.

Look here for a reference for all command line commands. The tilde-magic codes are described under for.

Yes, There are other shortcuts that you can use which are given below. In your command, ~d0 would mean the drive letter of the 0th argument.

~ expands the given variable
d gets the drive letter only
0 is the argument you are referencing

As the 0th argument is the script path, it gets the drive letter of the path for you. You can use the following shortcuts too.

%~1         - expands %1 removing any surrounding quotes (")
%~f1        - expands %1 to a fully qualified path name
%~d1        - expands %1 to a drive letter only
%~p1        - expands %1 to a path only
%~n1        - expands %1 to a file name only
%~x1        - expands %1 to a file extension only
%~s1        - expanded path contains short names only
%~a1        - expands %1 to file attributes
%~t1        - expands %1 to date/time of file
%~z1        - expands %1 to size of file
%~$PATH:1   - searches the directories listed in the PATH
               environment variable and expands %1 to the fully
               qualified name of the first one found.  If the
               environment variable name is not defined or the
               file is not found by the search, then this
               modifier expands to the empty string    

%~dp1       - expands %1 to a drive letter and path only
%~nx1       - expands %1 to a file name and extension only
%~dp$PATH:1 - searches the directories listed in the PATH
               environment variable for %1 and expands to the
               drive letter and path of the first one found.
%~ftza1     - expands %1 to a DIR like output line

This can be also found directly in command prompt when you run CALL /? or FOR /?

They are enhanced variable substitutions. They modify the %N variables used in batch files. Quite useful if you're into batch programming in Windows.

%~I         - expands %I removing any surrounding quotes ("")
%~fI        - expands %I to a fully qualified path name
%~dI        - expands %I to a drive letter only
%~pI        - expands %I to a path only
%~nI        - expands %I to a file name only
%~xI        - expands %I to a file extension only
%~sI        - expanded path contains short names only
%~aI        - expands %I to file attributes of file
%~tI        - expands %I to date/time of file
%~zI        - expands %I to size of file
%~$PATH:I   - searches the directories listed in the PATH
               environment variable and expands %I to the
               fully qualified name of the first one found.
               If the environment variable name is not
               defined or the file is not found by the
               search, then this modifier expands to the
               empty string

You can find the above by running FOR /?.