Q:

What did Jesus mean by saying 'to the right of the boat '?

We see at Luke 5:3-4  (NRSVCE)

He got into one of the boats, the one belonging to Simon, and asked him to put out a little way from the shore. Then he sat down and taught the crowds from the boat.   When he had finished speaking, he said to Simon, “Put out into the deep water and let down your nets for a catch.” 

We also see at John 21:3-6 (NRSVCE), in a different context, of course:

Simon Peter said to them, “I am going fishing.” They said to him, “We will go with you.” They went out and got into the boat, but that night they caught nothing.
   Just after daybreak, Jesus stood on the beach; but the disciples did not know that it was Jesus.   Jesus said to them, “Children, you have no fish, have you?” They answered him, “No.”   He said to them, “Cast the net to the right side of the boat, and you will find some.” 

One is unable to comprehend why Jesus asked the  disciples to cast the net to the right , when they would have done so hundreds of times while fishing 'throughout the night'. One is therefore, forced to assume that Jesus really meant 'into the deep water', as one sees at Luke 5:4 , by saying 'to the right'
My question therefore, is: Does the Catholic church teach that the terms 'to the right'and `into the deep water'as used by Jesus in two different contexts , mean one and the same . 

A:

I would argue that you are missing the significant point of the story.
See the next verse of the text you mentioned in John 21 

And He said to them, “Cast the net on the right side of the boat, and you will find some.” So they cast, and now they were not able to draw it in because of the multitude of fish.
    7 Therefore that disciple whom Jesus loved said to Peter, “It is the Lord!” Now when Simon Peter heard that it was the Lord, he put on his outer garment (for he had removed it), and plunged into the sea. 

This is Jesus revealing himself to the disciples. This is a callback to when he had original met Peter, which is the other account you had mentioned in (Luke 5)

But Simon answered and said to Him, “Master, we have toiled all night and caught nothing; nevertheless at Your word I will let down the net.” 6 And when they had done this, they caught a great number of fish, and their net was breaking. 7 So they signaled to their partners in the other boat to come and help them. And they came and filled both the boats, so that they began to sink. 8 When Simon Peter saw it, he fell down at Jesus’ knees, saying, “Depart from me, for I am a sinful man, O Lord!”

It's important to note that these are not two different accounts of the same event. The event in Luke happens when Peter and Jesus are first meeting each other, and the event in John happens after Jesus death and resurrection, ~ 3 years later. 
So there is no reason why the two phrases "To the right of the boat" would need to match "Put out into deep water". Because they are two different occurences, years apart. However, both with a similar point, revealing the Lord to his disciples, by showing his power over nature.

