Solution:
Let us compute $2010 * 10$ using the given properties.

Let $f(y) = 2010 * y$.

We are given:
$$f(0) = 0$$
$$f(y+1) = f(y) + (2010 - y)$$
Let us compute step by step:

$f(0) = 0$

$f(1) = f(0) + (2010 - 0) = 0 + 2010 = 2010$

$f(2) = f(1) + (2010 - 1) = 2010 + 2009 = 4019$

$f(3) = f(2) + (2010 - 2) = 4019 + 2008 = 6027$

$f(4) = f(3) + (2010 - 3) = 6027 + 2007 = 8034$

$f(5) = f(4) + (2010 - 4) = 8034 + 2006 = 10040$

$f(6) = f(5) + (2010 - 5) = 10040 + 2005 = 12045$

$f(7) = f(6) + (2010 - 6) = 12045 + 2004 = 14049$

$f(8) = f(7) + (2010 - 7) = 14049 + 2003 = 16052$

$f(9) = f(8) + (2010 - 8) = 16052 + 2002 = 18054$

$f(10) = f(9) + (2010 - 9) = 18054 + 2001 = 20055$

Therefore, $2010 * 10 = 20055$.

Alternatively, we can find a closed formula.

Let us try to find a general formula for $x * y$.

Let $f(y) = x * y$.

We have:
$$f(0) = 0$$
$$f(y+1) = f(y) + (x - y)$$
So,
$$f(1) = f(0) + (x - 0) = x$$
$$f(2) = f(1) + (x - 1) = x + (x - 1) = 2x - 1$$
$$f(3) = f(2) + (x - 2) = (2x - 1) + (x - 2) = 3x - 3$$
$$f(4) = f(3) + (x - 3) = (3x - 3) + (x - 3) = 4x - 6$$
So, the pattern is:
$$f(y) = yx - \frac{y(y-1)}{2}$$
Let us check for $y=4$:
$$f(4) = 4x - \frac{4 \times 3}{2} = 4x - 6$$
Correct.

Therefore,
$$2010 * 10 = 10 \times 2010 - \frac{10 \times 9}{2} = 20100 - 45 = 20055$$

Final answer:
$$\boxed{20055}$$