Create a Paint program!

8086 machine code, 32 31 28 26 bytes

00000000  b8 0f 00 cd 10 b8 02 10  ba 18 01 cd 10 b8 03 00  |................|
00000010  cd 33 4b b8 01 0c eb f3  3f 00                    |.3K.....?.|
0000001a

Assumes the presence of a VGA graphics card, and any Microsoft-compatible mouse driver.

-2 bytes thanks to @berendi!

How it works:

            |   org 0x100
            |   use16
b8 0f 00    |       mov ax, 0x000f      ; set screen mode: 640x350 monochrome
cd 10       |       int 0x10            ; call video bios
b8 02 10    |       mov ax, 0x1002      ; set palette
ba 18 01    |       mov dx, palette     ; pointer to palette data
cd 10       |   @@: int 0x10            ; call video bios
b8 03 00    |       mov ax, 0x0003      ; get mouse position in (CX, DX) and button status in BL
cd 33       |       int 0x33            ; call mouse driver
4b          |       dec bx              ; if no buttons pressed, --BX = 0xFFFF (invalid page)
b8 01 0c    |       mov ax, 0x0c01      ; plot pixel with color AL at (CX, DX) in page BH
eb f3       |       jmp @b              ; repeat
3f 00       |   palette db 0x3f, 0x00   ; palette in 2:2:2 rgb. color 0 = white, 1 = black.
            |                           ; this should be a 17-byte struct, but we only care about the first two colors here

That's all there is to it. No magic involved, just a few function calls.

Scratch, 10 Scratch bytes

Scratch was practically designed for things like this.

Try it online!

HTML + JavaScript (ES6), 66 64 62 bytes

HTML

<canvas id=c>

JavaScript

onclick=e=>c.getContext`2d`.fillRect(e.x,e.y,1,1)

Saved 2 bytes thanks to Gustavo Rodrigues and 2 bytes from Shaggy.

onclick=e=>c.getContext`2d`.fillRect(e.x,e.y,1,1)

/* This css isn't needed for the program to work. */
*{margin: 0}
#c{border:1px solid black}

<canvas id=c>