FIR filter design and implementation meaning
I don't fully understand how you are expressing your question in English, so please accept my attempt.

Discrete convolution is 
$$ y[n]=\sum_{k=-\infty}^{\infty} x[n-k] h[k]
$$
where $y[n]$ is the output at time index $n$ and $h[k]$ is the impulse response.  The index $k$ ranges over $-\infty$ to $\infty$ if $h[n]$ is an infinite impulse response.

A causal convolution, i.e. the filter can only have output $y[n]$ for $x[n]$ ,$x[n-1]$ ,$x[n-2]$ $\dots$ $x[n-\infty]$ 
$$ y[n]=\sum_{k=0}^{\infty} x[n-k] h[k]
$$
if $h[0]\ne 0$, $h[1]\ne 0$, $h[2] \ne 0$, $\dots$ $h[N-1]\ne 0$ , $h[N]=0$,  $h[N+1]=0$, $\dots$,  $h[\infty]=0$, is of finite length, it is a FIR filter and


  what is the output of the convolution meaning which is used in the FIR
  filter ?


$$ y[n]=\sum_{k=0}^{N-1} x[n-k] h[k]
$$


  how can conclude the output of the FIR filter in the output according
  to the input x[n]?


$y[n]$ depends on a finite sum of $x[n]$  .  A FIR filter also need not necessarily be causal, but it doesn't change that $y[n]$ only depends on a finite sum of $x[n]$ 

I don't understand how design has anything to do with your question.