Why are LTI systems defined by convolution, why not in any other way?
it takes more work to do this for continuous-time systems, so i'll just do it for discrete-time and we can all wave our arms to extend to continuous-time.

a general discrete-time input signal can be expressed as

$$ x[n] = \sum\limits_{i=-\infty}^{\infty} x[i] \delta[n-i] $$

where $\delta[n]$ is the Kronecker delta or "unit impulse function"

$$ \delta[n] \triangleq
\begin{cases}
1, &amp; \text{if }n = 0 \\
0, &amp; \text{if }n \ne 0
\end{cases} $$

so think of your input $x[n]$ as the sum of a bunch of individual impulse functions, each delayed to time $i$ and weighted by a constant which is equal to $x[i]$.

so, let's say we have a Linear and Time-Invariant discrete-time system, represented with operator, $\mathbb{T}$, and output:

$$ y[n] = \mathbb{T}\left\{ x[n] \right\} $$

and we input into that system a single unit impulse, $\delta[n]$.  by definition, the "impulse response" of the LTI system is the output coming out when a unit impulse, $\delta[n]$, is input.

$$ h[n] \triangleq \mathbb{T}\left\{ \delta[n] \right\} $$

now, imagine a general input, $x[n]$ is input to $\mathbb{T}\{ \}$.  then, because it's Linear:

$$ \begin{align}
y[n] &amp; = \mathbb{T}\left\{ x[n] \right\} \\
&amp; = \mathbb{T}\left\{ \sum\limits_{i=-\infty}^{\infty} x[i] \delta[n-i] \right\} \\
&amp; = \sum\limits_{i=-\infty}^{\infty} \mathbb{T}\left\{ x[i] \delta[n-i] \right\} \\
&amp; = \sum\limits_{i=-\infty}^{\infty} x[i]  \mathbb{T}\left\{ \delta[n-i] \right\} \\
\end{align} $$

and because it's Time Invariant, 

$$ h[n-i] = \mathbb{T}\left\{ \delta[n-i] \right\} $$

then, putting the two together, you get

$$ \begin{align}
y[n] &amp; = \sum\limits_{i=-\infty}^{\infty} x[i]  \mathbb{T}\left\{ \delta[n-i] \right\} \\
&amp; = \sum\limits_{i=-\infty}^{\infty} x[i]  h[n-i] \\
\end{align} $$