MCO 15 - need an explanation of terminology please

To hold the pawn basically means make every effort to keep it and maintain having this extra pawn, even if this makes your position slightly awkward and you have to perform some contortions.

To return the pawn means that you do not attempt to keep this extra pawn and instead develop pieces, improve your position in other ways. 

It comes down to the validity of the gambit. One player gambits a pawn and says to the other 'I don't care about this pawn, by giving it away I get x y and z as compensation and it's well worth it'. The other player must then decide for himself if this is correct. Is it worth trying to keep hold of the extra pawn or do you have to make too many harmful positional concessions to do that? Is it safer or better to forget trying to keep hold of it and focus on position rather than material? 

What you describe is 'accepting' or 'declining' the gambit. 1.e4 e5 2.f4 exf4 is accepting, 1.e4 e5 2.f4 Nc6 is declining. The holding or returning describes what takes place after the gambit has been accepted.

So in the Kings gambit accepted...

Black now has 2 strategies. He can either try to hold on to his extra pawn on f4 or he can return it.

For example, holding it like this:

These are just some moves to demonstrate the idea, there's all sorts of ways to play. But you see that black has solidly defended the pawn but white is ahead in development and black has weakened his kingside. Both sides are arguing that their strategy will pay off - white that the development lead will allow him to create threats and gain an advantage, black that he can defend against these threats and that the pawn will prove more important.

Black can also instead return the pawn like this:

Again just some moves to demonstrate the idea. Black has made no effort to hold on to the f4 pawn and has thus returned it to white. In the meantime he has developed his kingside pieces, castled, and retained a solid structure. Black prefers not to get into the complications which arise from trying to hold the extra pawn and instead has a simpler, safer position but with equal material.

Which strategy you choose is largely a matter of preference and knowledge. 

Thank you! I'll print this off and study. I have so much to learn! I really appreciate the time taken to respond in depth. Best wishes. Bill 

You're welcome Bill. I see on your profile you're very new to chess so if there's other stuff in future you're not sure about or whatever feel free to ping me a message and I'll try to help 👍. There are layers and layers of complexity and also terminology actually so try not to drown in it! 

Thanks Andy. Much appreciated. Not doing so very well in my gameplay at the moment as you can see. Very willing to learn though.