I don't understand this

what is the modular inverse of 11 , modulo 1000?

The answer =91, because:

[91 x 11] mod 1000 = 1, which is the definition of "modular multiplicative inverse"

Let $x \equiv 11^{-1}\pmod{1000}$

Now, because $11^{\varphi(1000)} \equiv 1 \pmod{1000}$, where $\varphi(x)$ is the Euler totient function,

$x\equiv 11^{\varphi(1000) - 1} \pmod{1000}$

We proceed to find the value of $\varphi(1000)$.

$\varphi(1000) = 1000\left(1 - \dfrac12\right)\left(1 - \dfrac15\right) = 400$

Therefore

$x\equiv 11^{399} \pmod{1000}$

$x\equiv 1331^{133} \pmod{1000}\\ x\equiv 331^{133}\pmod{1000}$

Now, 

$331^{133}\equiv 3 \pmod 8$

$331^{133}\equiv 81^{33}\equiv3^{32} = 7^6 \cdot 9 \equiv 9216 \equiv 91\pmod{125}$

By CRT, $x \equiv 331^{133}\equiv 91 \pmod{1000}$

This means $11^{-1} \equiv 91 \pmod{1000}$.