curl

curl - command line tool for sending web requests.

Windows - can be installed with Scoop or Chocolatey.
Linux - can be installed with built-in package manager or Homebrew.
macOS - can be installed with Homebrew or MacPorts.

Also, you can download executable from their website and run it from a directory or make it discoverable through PATH, so it can be run from anywhere. Using a package manager is preferable as it simplifies the process of installing and updating the tool.

Usage

curl -H "Content-Type: application/json" -d <body> <link>

Bash/Zsh/etc.

# -H "Content-Type: application/json" - adds header that tells server you're sending JSON data.
# -d '{"username": "test", "content": "hello"}' - sets request data.
# Using -d also sets method to POST, so -X POST can be ommited.
curl -H "Content-Type: application/json" -d '{"username": "test", "content": "hello"}' "https://discord.com/api/webhooks/123/w3bh00k_t0k3n"

# To make command more readable you can split it to multiple lines using backslash `\`
# and set webhook url as variable, so you don't need to paste it over and over again.
# Also you can add the variable to your `.*rc` file, so it persists on console reloads.
export WEBHOOK_URL="https://discord.com/api/webhooks/123/w3bh00k_t0k3n"
curl \
  -H "Content-Type: application/json" \
  -d '{"username": "test", "content": "hello"}' \
  $WEBHOOK_URL

PowerShell

# In PowerShell you have to escape " with \ inside strings, so body string be parsed correctly.
$WEBHOOK_URL = "https://discord.com/api/webhooks/123/w3bh00k_t0k3n"
curl -H "Content-Type: application/json" -d '{\"username\": \"test\", \"content\": \"hello\"}' $WEBHOOK_URL

# ` is used for making command multiline
curl `
  -H "Content-Type: application/json" `
  -d '{\"username\": \"test\", \"content\": \"hello\"}' `
  $WEBHOOK_URL

Command Prompt (cmd.exe)

REM Notice double quotes around body and none around link.
SET WEBHOOK_URL=https://discord.com/api/webhooks/123/w3bh00k_t0k3n
curl -H "Content-Type: application/json" -d "{\"username\": \"test\", \"content\":\"hello\"}" %WEBHOOK_URL%

REM ^ is multiline splitter in cmd.
curl ^
  -H "Content-Type: application/json" ^
  -d "{\"username\": \"test\", \"content\": \"hello\"}" ^
  %WEBHOOK_URL%

Sending attachments

# -F 'payload_json={}' - sets json body.
# -F "file1=@cat.jpg" - adds cat.jpg file as attachment.
# -F "file2=@images/dog.jpg" - adds dog.jpg file from images directory.
# Using -F also sets "Content-Type: multipart/form-data" header specifying it can be ommited.
curl \
  -F 'payload_json={"username": "test", "content": "hello"}' \
  -F "file1=@cat.jpg" \
  -F "file2=@images/dog.jpg" \
  $WEBHOOK_URL