Conceptual questions from signal processing
Batman has given a great answer. You need to go through the recommended book in order to understand the concepts mentioned. Let me try to simplify it. 

BIG PICTURE: De-convolution or inverse filtering is required to retrieve an estimate of the original signal that went through an unknown linear system. Basically, we have a signal which went through an unknown system and we want to get the original back. 
Example: You shouted something across a room with lot of reverberation (echos) to your brother. Now for your brother to understand it, he needs to remove the reverberation effect. Let us see it in signal processing terminology.    

What is convolution: Let $x(k)$ be your speech signal, and $h(k)$ is the unknown response of the room, then speech heard by your brother is $y(k) = h(k)\ast x(k)$ (this is convolution) and is more formally defined as:

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

Basically, your brother is listening to attenuated copies of your sound, i.e., $y(k) = h(0)x(k) + h(1)x(k-1) + ... + h(N-1)x(k-N+1)$. Note here x(k) is the complete speech of yours and $k-i$ denotes shift by $i$th sample, referring to the shifted copies.   

What is deconvolution: Now lets say your brother recorded $y(k)$ and he wants to know what did you say? i.e., what is $x(k)$. What he needs to do is design a filter $w(k)$ such that $\delta(k) = w(k)\ast h(k)$, i.e., $w(k)$ is the inverse of $h(k)$. If you know basic signal processing then $\delta$ is the dirac delta function or an impulse. Now if we convolve $y(k)$ with $w(k)$ then $w(k)\ast y(k) = w(k)\ast h(k) \ast x(k) = \delta(k) \ast x(k) = x(k)$. And he can know what you said.  

Minimum Phase: deals with the design of $w(k)$. Minimum phase also means that our filter is designed such that it responds much faster than another filter that is non-minimum phase. It has to do with setting of zeros of $w(k)$. This concept will require another discussion, which I'll postpone for now.