DOS keeps one current drive, and every file or directory name without a drive letter refers to it. INT 21h AH=0Eh changes it, which is what the shell does when you type Z: at the prompt. The drive in DL counts from 0: A = 0, B = 1, C = 2, the same numbering INT 21h AH=19h returns. The services that ask about a drive, such as INT 21h AH=36h and INT 21h AH=47h, count from 1 instead and use 0 for the current drive.

There is no error return. Given a drive that does not exist, DOS leaves the current drive as it was and returns AL as usual. To know whether the switch worked, ask with 19h afterwards:

MOV     AH, 0Eh  ; try drive D:
MOV     DL, 3
INT     21h
MOV     AH, 19h  ; where are we now?
INT     21h
CMP     AL, 3
JNE     NODRIVE  ; still on the old drive: there is no D:

AL comes back with the number of drive letters, not the number of drives. MS-DOS returns the LASTDRIVE setting of CONFIG.SYS (5, A to E, unless set), or the number of block devices if that is larger. DOSBox-X returns one past the highest mounted letter, and since its own drive Z: is always there, the answer is 26.

prints: C: -> Z: -> C:, drive letters: 26 (when the program starts on C:)
MOV     AH, 19h       ; DOS: the current drive in AL, 0 = A
INT     21h
MOV     OLD1, AL      ; to come back to
CALL    PDRV
MOV     AH, 0Eh       ; DOS: select drive DL
MOV     DL, 25        ; 25 = Z:, DOSBox-X's own drive
INT     21h
MOV     COUNT1, AL    ; AL = the number of drive letters
MOV     DX, OFFSET ARROW1
MOV     AH, 09h
INT     21h
MOV     AH, 19h       ; did it work?
INT     21h
CALL    PDRV
MOV     AH, 0Eh       ; back to the old drive
MOV     DL, OLD1
INT     21h
MOV     DX, OFFSET ARROW1
MOV     AH, 09h
INT     21h
MOV     AH, 19h
INT     21h
CALL    PDRV
MOV     DX, OFFSET COUNTM1
MOV     AH, 09h
INT     21h
MOV     AL, COUNT1
MOV     AH, 0
CALL    PDEC
MOV     DX, OFFSET CRLF
MOV     AH, 09h
INT     21h
 
OLD1    DB      0
COUNT1  DB      0
ARROW1  DB      " -> $"
COUNTM1 DB      ", drive letters: $"
CRLF    DB      13, 10, "$"
PDRV    PROC    NEAR  ; print drive AL (0 = A) as "C:"
ADD     AL, "A"
MOV     DL, AL
MOV     AH, 02h
INT     21h
MOV     DL, ":"
MOV     AH, 02h
INT     21h
RET
PDRV    ENDP
PDEC    PROC    NEAR  ; print AX in decimal (changes AX BX CX DX)
MOV     BX, 10
XOR     CX, CX        ; digits on the stack
PDEC1:
XOR     DX, DX
DIV     BX            ; AX = AX / 10, DX = the last digit
PUSH    DX
INC     CX
OR      AX, AX
JNZ     PDEC1
PDEC2:
POP     DX            ; the digits come back first to last
ADD     DL, "0"
MOV     AH, 02h
INT     21h
LOOP    PDEC2
RET
PDEC    ENDP

Every drive has its own current directory

DOS remembers a current directory for each drive. Switching to Z: and back to C: lands in the same C: directory as before, and INT 21h AH=3Bh with a path such as D:\GAMES changes the directory of D: without making D: current. The current drive is not part of the program either: a program that switches drives and exits leaves the user on the new drive, so put the old one back first, as the example does.

Floppies

On a PC with one floppy drive, A: and B: are the same physical drive. Selecting B: works, and the first access to it makes DOS ask the user to insert the diskette for drive B:, and to swap again when A: is used. A program that walks through the drives to find disks should use INT 21h AH=36h or an IOCTL call to check each one, not just select it.