there are two components to "dealing" with a dog's aggression, both are equally important:
1. Management - this means physically preventing the behavior from starting in the first place so that you won't have an incident on your hands. Examples of preventing the behavior include just keeping the dog away from other dogs, or muzzling the dog when around other dogs. Management doesn't make the dog any less aggressive so it's not curing the dog, but it is necessary to prevent injuries or prevent the aggression from worsening. Management also needs to be done while the dog is undergoing a rehabilitation program so as to prevent setbacks to the rehabilitation process. And, for most if not all dogs who have aggression issues, management should be a lifelong thing as well even if they have made great strides towards rehabilitation.
2. Rehabilitating the dog, treating his underlying cause of aggression. Also called "behavior modification." The goal here is to re-program the dog's brain to feel less aggressive in the first place, think of it as being like psychiatric treatment for dogs! However note that being "cured" is a vague term and not all dogs can in fact be fully cured or even substantially cured. However, with the proper treatment they can certainly get better meaning they can become more tolerant and less aggressive than they were before. And any improvement helps because it lessens the management burden just a little or increases your margin of safety during management.
It sounds like your trainer is talking about mainly about management...?
Some things that fall under management include:
1. Obedience training - training the dog to make eye contact with you and to hold eye contact, can be very useful. Because then you command the dog to pay attention to you, and if he is paying attention to you he can't also be watching and concentrating on the other dog. However, there are limits to how effective and appropriate this technique is, e.g. if the other dog is right up against yours, you can't expect him to maintain eye contact! this is a good and useful thing to get through quick and low-intensity situations before your dog has started getting worked up. e.g. letting another dog and owner pass you by in the distance. It will probably fail under high stress situations e.g. if your dog is already going over the top reacting at another dog...
you can also use "leave it" command too, same thing, you are training the dog to divert his attention away on command. But you don't want to have your dog do this IF something is going to happen that will stress him out so completely he fails the exercise, because then this can hurt his trust in you and make it harder to use this technique in future. so using the dog's obedience training to control his behavior and prevent situations from escalating is appropriate for a lot but not all situations.
But the good news is that the more gradual and successful/positive practice sessions you and your dog do in this, the easier it will get and the more the dog will be able to do it in future. it's really about the dog's trust - he is trusting that when you tell him to do something in a stressful situation that he can do it and nothing bad (from the other dog) will happen to him.
2. Muzzling - this is always the safest route and gives you the most peace of mind. However it may make other people react to you in a negative way because a muzzle looks like you're saying your dog is Cujo! Many trainers will muzzle the dog (and the dog is gradually acclimatized to the muzzle so he doesn't mind wearing it) while rehabilitating, just for the safety of everyone involved.
3. Some trainers recommend using a Gentle Leader or Halti head collar on the dog, this is so that you can gently manipulate your dog's head to be looking away from whatever he would otherwise fixate on. Again, preventing the dog from staring and obsessing at the other dog helps to prevent him from getting worked up and buys you time to let the other dog pass by without incident or you can take your dog away from the situation.
4. Just avoiding situations where your dog can come into contact with other dogs, such as walking at different times of the day or choosing less crowded areas to go to. this doesn't solve the problem, but it certainly prevents it getting worse....
As for rehabilitating the aggression, you would probably be best working one on one with a trainer or behaviorist. I like the book Sammy recommended (Click to Calm by Emma Parsons) and also what Sammy suggested. Here are a couple more books along the same lines: "Cautious Canine" by Patricia McConnell, "Feisty Fido" also by Patricia McConnell, "Scaredy Dog!" by Ali Brown.
The reason for the titles of these books is because often aggression has an underlying fear component (the dog is uncomfortable around other dogs, if he were comfortable he wouldn't be reacting). The counterconditioning process is a long-term process and can be tedious, it's all about taking baby steps where you expose the dog to other dogs (but in a highly controlled setting) and make it a positive experience.
So for example with your daughter's mastiff, one thing you could do is set up a controlled session where the mastiff is either on leash or safely behind a fence or your dog is muzzled (these are precautions for safety in case your dog lunges and bites). Then you bring your dog on leash to a distance where she sees the mastiff but is not yet stressed or bothered much, you guys should be far enough away from the mastiff that your dog can still listen to you and take treats gently from your hand. Chomping down on your hand when taking treats is a sign of being worked up and that you should probably back up some more. Then just feed steady stream of really good food while your dog is looking at the mastiff. Then end the session completely. next session repeat and try to go a couple steps closer to the mastiff. Or just stay at the same level (distance), whatever keeps your dog comfortable enough that she can see the mastiff yet remain calm and be able to be happy about the really good treats. each dog progresses at a different rate. eventually - if there are no setbacks in between sessions - your dog should gradually get more comfortable and be able to go closer to the mastiff without getting so stressed. but again, how much and how fast the dog will progress is highly individual to the dog and your specific situations so it's much better to be working with a trainer or behaviorist so they can give you a better idea and plan.
In between sessions, during 'real life', to prevent setbacks to the counterconditioning you should prevent her from getting into situations where she gets upset at the mastiff or at other dogs. this is because during the sessions she's learning it's OK to get closer to other dogs, so if in between sessions she gets upset all over again, enough to lunge and bite, that sort of cancels out whatever the sessions accomplished.... So in between sessions, the "daily life' part, this is where the management comes in.
it's really better and safer to work with a trainer or behaviorist because aggression is a serious issue that needs experienced professionals to guide you. Good luck!!