What should a reply to a peer review look like?
Generally, the best way to respond is with a detailed rundown, discussing every point that the referee raised in turn.  (This may be unnecessary if there are only a handful of minor changes asked for, but it doesn't hurt even then.)

What I generally do is to start with a two or three sentence introduction, stating that I have made all the changes that the referee suggests.  If there were changes that I was unwilling to make, I just say that I have made most of the changes but that I feel that one or more of the referee's requests were not appropriate.

Then I begin a point-by-point analysis of what the referee said and how I have responded.  Usually, I address the points in the same order as the referee raised them in their report, but not always.  (Sometimes there may be just one major revision that was requested, and I want to discuss that first, before the minor points, even though the referee did not order things the same way.)  It's often a good idea to quote directly what the referee said in the report and then follow up with an explanation of the changes that have been made.

Sometimes, just stating that the change has been made is sufficient.  This is typically all that's needed for linguistic problems.  If the editor and referee may have trouble finding a change, you can point out where it is.  If the changes are more substantive, it's a good idea to give a summary of what the changes were.  These don't need to be long, but the should point out the main significance of the changes.

If there were changes that you were unwilling to make, you must carefully explain why.  If you think the referee was wrong, this should be pointed out very tactfully.  A good referee will recognize their error if they really made one.  If there are matters of opinion, it may be trickier to convince editor and referee that things are really better your way.  It is also important that you make it clear that you are reasonable about making changes.  If you refuse to make anything but the most trivial corrections in response to a referee's report, that makes you look stubborn and probably unreasonable.