INT 21h AH=3Bh sets the current directory: CD (CHDIR) at the prompt. A relative path starts from the current directory, a path starting with "\" from the root, and ".." means the parent:

PathResult
SUBinto SUB, below the current directory
..up one level
\the root of the current drive
\TOOLS\BINan absolute path on the current drive
D:\GAMESthe current directory of D:; the current drive does not change

The drive in the path does not become the current drive; that takes INT 21h AH=0Eh. This is why CD D:\GAMES at a C:\> prompt leaves you on C:. To go somewhere on another drive, a program makes both calls.

Pitfalls: a trailing backslash (\TOOLS\) fails with error 3, except for the root itself; the path must be ASCIIZ, and name a directory, not a file. The whole path, drive included, must fit in 64 characters.

Coming back

The current directory belongs to the drive, not to the program. DOS keeps one per drive in its current directory structure, and a program that changes it and exits leaves the user there. Save it with INT 21h AH=47h first and change back before exiting. 47h returns the path without the leading backslash, so the example keeps a "\" in front of its buffer to make the saved path absolute:

prints: C:\ / back in the old directory
MOV     AH, 47h            ; DOS: the current directory, for later
MOV     DL, 0              ; 0 = the current drive
MOV     SI, OFFSET OLD1+1  ; after the "\" already there
INT     21h
MOV     AH, 3Bh            ; DOS: change the directory to DS:DX
MOV     DX, OFFSET ROOT1   ; "\": the root
INT     21h
CALL    PCWD               ; prints "C:\"
MOV     AH, 3Bh            ; and back: "\" and the old path
MOV     DX, OFFSET OLD1
INT     21h
MOV     AH, 47h            ; where are we now?
MOV     DL, 0
MOV     SI, OFFSET NOW1
INT     21h
CLD
MOV     SI, OFFSET OLD1+1
MOV     DI, OFFSET NOW1    ; ES = DS in a .COM
MOV     CX, 64
REPE    CMPSB              ; the same 64 bytes?
MOV     DX, OFFSET BACK1
JE      SAME1
MOV     DX, OFFSET LOST1
SAME1:
MOV     AH, 09h
INT     21h
 
ROOT1   DB      "\", 0
OLD1    DB      "\", 64 DUP(0)
NOW1    DB      64 DUP(0)
BACK1   DB      "back in the old directory", 13, 10, "$"
LOST1   DB      "somewhere else", 13, 10, "$"
PCWD    PROC    NEAR       ; print the current directory: C:\PATH
MOV     AH, 19h            ; the drive letter
INT     21h
ADD     AL, "A"
MOV     DL, AL
MOV     AH, 02h
INT     21h
MOV     DL, ":"
MOV     AH, 02h
INT     21h
MOV     DL, "\"
MOV     AH, 02h
INT     21h
MOV     AH, 47h            ; the directory, without "\" in front
MOV     DL, 0
MOV     SI, OFFSET DIRBUF
INT     21h
CLD
PCWD1:
LODSB
OR      AL, AL
JZ      PCWD2
MOV     DL, AL
MOV     AH, 02h
INT     21h
JMP     PCWD1
PCWD2:
MOV     DX, OFFSET CRLF
MOV     AH, 09h
INT     21h
RET
PCWD    ENDP
DIRBUF  DB      64 DUP(0)
CRLF    DB      13, 10, "$"