Understanding the Difference Between MAP Estimation and ML Estimation
You have a set of message set $m_i$, $0 \le i \le N-1$. (For example, QPSK will be $N=4$). For the transmitted message $m_i$, the corresponding symbol vector is $\textbf{x}_i$, and the received symbol vector is $\textbf{y} = \textbf{x} + \textbf{w}$, where $\textbf{w}$ is the AWGN at the receiver. The above is a simplified baseband model assuming a simple Line-Of-Sight(LOS) channel without any delay.

At the receiver, after observing $\textbf{y}$ you arrive at a decision for the transmitted symbols, $\tilde{\textbf{x}}_i$. The decision you make is such that the probability of error $P_e = P(\tilde{m_i} \ne m_i)$ is minimum. In other words, the Probability of being correct $P_c = 1 - P_e$ has to be maximized. Using a hypothetical rule you have made a decision $\tilde{m}_i$ based on $\textbf{y}$, so 
$$
P_c(\tilde{m} = m_i;\textbf{y}) = P(\tilde{\textbf{x}} = \textbf{x}_i;\textbf{y}) = 
P(\textbf{x}_i|\textbf{y})P(\textbf{y})
$$
Here I have made use of formula $P(AB) = P(A|B)P(B)$. Event A is the event that $x_i$ was transmitted. Our decision is correct if $\tilde{x_i}$ is interpreted as $x_i$. We are trying to maximize the probability of getting this decision correct. In order to maximize the above term among all message set $0 \le i \le N-1$, the term $P(\textbf{y})$ can be ignored since it does not depend on $i$. Hence 

$$
\tilde{m_i} = argmax_i\,\, P(\textbf{x}_i|\textbf{y})
$$
You are deciding the $i$ which maximizes the above probability after observing $\textbf{y}$. Hence the whole method above is called Maximum A-posteriori Probability decision because you are maximizing a-posteriori probability $P(\textbf{x}_i|\textbf{y})P(\textbf{y})$

The term above can be re-written as $P(\textbf{x}_i|\textbf{y}) = P(\textbf{y}|\textbf{x}_i)P(\textbf{x}_i)/P(\textbf{y})$. 

If we assume equal probability for all messages in the set $\textbf{m}$ then we can ignore the term $P(\textbf{x}_i)$ since they are equal for all values of $i$. Hence
$$
\tilde{m_i} = argmax_i\,\, P(\textbf{y}|\textbf{x}_i)
$$
This is the Maximum Likelihood detection rule. We are maximizing the likelihood probability $P(\textbf{y}|\textbf{x}_i)$.