Have just complained today that dons expose the topic very vaguely. I advise you to read that along with the glance at time diagram. Basically, if your question is not about Matlab, input response is a way you can compute response of your system, given input $\vec x = [x_0, x_1, x_2, \ldots x_t \ldots]$. That is a vector with a signal value at every moment of time. That is a waveform (or PCM encoding) of your known signal and you want to know what is response $\vec y = [y_0, y_2, y_3, \ldots y_t \ldots]$. This is a vector of unknown components. Again, every component specifies output signal value at time t. The idea is that you can compute $\vec y$ if you know the response of the system for a couple of test signals and how your input signal is composed of these test signals. You should be able to expand your $\vec x$ into a sum of test signals (aka basis vectors, as they are called in Linear Algebra).
I advise you to look at Linear Algebra course which teaches that every vector can be represented in terms of some chosen basis vectors $\vec x_{in} = a\,\vec b_0 + b\,\vec b_1 + c\, \vec b_2 + \ldots$. Here, a is amount of vector $\vec b_0$ in your signal, b is amount of vector $\vec b_1$ in your signal and so on. There is noting more in your signal. It is just a weighted sum of these basis signals. You may call the coefficients [a, b, c, ..] the "specturm" of your signal (although this word is reserved for a special, fourier/frequency basis), so $[a, b, c, ...]$ are just coordinates of your signal in basis $[\vec b_0 \vec b_1 \vec b_2]$. The point is that the systems are just "matrices" that transform applied vectors into the others, like functions transform input value into output value. once you have measured response of your system to every $\vec b_i$, you know the response of the system for your $\vec x.$ That is it, by virtue of system linearity. That is, suppose that you know (by measurement or system definition) that system maps $\vec b_i$ to $\vec e_i$. Your output will then be $\vec x_{out} = a \vec e_0 + b \vec e_1 + \ldots$! Voila! However, the impulse response is even greater than that. It allows to know every $\vec e_i$ once you determine response for nothing more but $\vec b_0$ alone!
The basis vectors for impulse response are $\vec b_0 = [1 0 0 0 ...], \vec b_1= [0 1 0 0 ...], \vec b_2 [0 0 1 0 0...]$ and etc. That is, your vector [a b c d e ...] means that you have a of [1 0 0 0 0] (a pulse of height a at time 0), b of [0 1 0 0 0 ...] (pulse of height b at time 1) and so on. The system system response to the reference impulse function $\vec b_0 = [1 0 0 0 0]$ (aka $\delta$-function) is known as $\vec h = [h_0 h_1 h_2 \ldots]$. It looks like a short onset, followed by infinite (excluding FIR filters) decay.
This is a picture I advised you to study in the convolution reference. Here is why you do convolution to find the output using the response characteristic $\vec h.$ As you see, it is a vector, the waveform, likewise your input $\vec x$. These signals both have a value at every time index. The reaction of the system, $h$, to the single pulse means that it will respond with $[x_0, h_0, x_0 h_1, x_0 h_2, \ldots] = x_0 [h_0, h_1, h_2, ...] = x_0 \vec h$ when you apply the first pulse of your signal $\vec x = [x_0, x_1, x_2, \ldots]$. The first component of response is the output at time 0, $y_0 = h_0\, x_0$. The rest of the response vector is contribution for the future. You will apply other input pulses in the future. They will produce other response waveforms. But, the system keeps the past waveforms in mind and they add up. That is, at time 1, you apply the next input pulse, $x_1$. It will produce another response, $x_1 [h_0, h_1, h_2, ...]$. The output at time 1 is however a sum of current response, $y_1 = x_1 h_0$ and previous one $x_0 h_1$. The output of a signal at time t will be the integral of responses of all input pulses applied to the system so far, $y_t = \sum_0 {x_i \cdot h_{t-i}}.$ That is a convolution. This operation must stand for . in your example (you are right that convolving with const-1 would reproduce x(n) but seem to confuse zero series 10000... with identity 111111..., impulse function with impulse response and Impulse(0) with Impulse(n) there). I have told you that [1,0,0,0,0..] provides info about responses to all other basis vectors, e.g. [0,1,0,0,0,...], because shifted (time-delayed) input implies shifted (time-delayed) output.
Actually, frequency domain is more natural for the convolution, if you read about eigenvectors. However, because pulse in time domain is a constant 1 over all frequencies in the spectrum domain (and vice-versa), determined the system response to a single pulse, gives you the frequency response for all frequencies (frequencies, aka sine/consine or complex exponentials are the alternative basis functions, natural for convolution operator). Basically, it costs t multiplications to compute a single components of output vector and $t^2/2$ to compute the whole output vector. In the frequency domain, by virtue of eigenbasis, you obtain the response by simply pairwise multiplying the spectrum of your input signal, X(W), with frequency spectrum of the system impulse response H(W).
I hope, this will make things clear.
