Imagine that you have a value response that could be a few things:

response = {:ok, result}
response = {:error, reason}
response = {:invalid, :other, :fields}

Then the cases would be:

# This would work for the first two cases returning result in the first one,
# {:error, other} for the second one and raise an exception for the third one (no match at all).
case response do
  {:ok, result} -> result
  {_, other} -> {:error, other}
end

# This would work for all the cases and return result for the first case,
# {:error, other} for the second case and
# {:error, {:invalid, :other, :fields}} for the third one
case response do
  {:ok, result} -> result
  other -> {:error, other}
end

# This would return result for the first case and :ok for any other response.
case response do
  {:ok, result} -> result
  _other -> :ok
end

Usually you will match a value that comes from a function and the clauses of the match will depend of the format of the result of that function.