The modulation and coding techniques are characterized by what is called bit error rate BER versus the signal to noise and interference ratio. Every modullation and channel coding technique has a specific BER versus energy per bit to noise density ratio. These curves are used to determine the minimum required S/N density ratio for certain quality of service expressed in an upper bound of bit error rate.
As for the the CQI it is the channel quality indicator which is the measured S/N ratio at the receiver under specific channel condition and at specific time. It is used to adopt the coding and modulation to the channel state.
MCS is the modulation and coding scheme that the eNB and UE agreed on.
The CQI is a 4-bit integer and is based on the received signal-to-interference-plus-noise ratio (SINR) at the User Equipment (UE) .
CQI is sent from UE to the eNode-B (UP-link) to indicate a suitable downlink transmission data rate, i.e., (((certain MCS value ))).
Therefore, CQI is an indicator carrying the information on how good/bad the communication channel quality is.
In LTE networks, there are 15 different CQI values ranging from 1 to 15 and mapping between CQI and modulation scheme (see attached table for more details)