Whether you are introducing dogs, puppies, males, or females, the scenario is usually best if both dogs can meet on neutral ground, off-lead.
You will have a better idea of your dog's behavior, and more control of your dog if he knows a few basic commands. Same thing for the other dog. I know it is not usually common to have the opportunity to train a dog prior to entering your household, but if you have that opportunity, take advantage of it. Otherwise, begin obedience training with your new dog on his first day of arrival.
If you are adopting a dog from a humane society or responsible breeder or rescue organization, they should have no problem with allowing your dog to accompany you for the final phase of the selection process. Pick two or three dogs that you think would be compatible, and then let your own dog see who he likes as well. The dogs should meet in an outside fenced area off-lead unless your (or the other dog) has aggressive tendencies.
If you are not adopting a new dog, but just want your dog to meet a new dog an open park or some other neutral territory that is relatively new to both dogs will be a good meeting ground. If you choose your dog's favorite park, he may react territorially if he's not used to seeing other dogs at "his" park.
Before letting the dogs see each other, or get too close, you may want to have pre-arranged to have a cloth or small towel with the other dogs scent on it. (The other dog can be presented with a similar item from his handler, with your dog's scent on it.) As you let the dog sniff the cloth talk in a happy voice and say the new dog's name. "Can you smell Shaggy?" "You're going to meet Shaggy today!" Give your dog a few good food treats and praise him saying, "Rover likes Shaggy!" "Good Say Hello to Shaggy!" Then play with him for a few minutes. The idea is to associate the new dog's smell with good things: upbeat voice, food, praise, and play. If you're walking to a park, you can do this exercise two or three times before reaching your destination. If you're at a shelter your dog may be too nervous or distracted to play or do more than once.
Next, the ideal meeting will be two unleashed dogs in an enclosed fenced area unaware of each other. If they can meet each other naturally, on their own time it usually works best. If the enclosed area is too small for them not to notice each other, but has enough space that it is not crowded, bring them both in the area on opposite sides, on lead, heeling. Place them in a sit and do a few obedience commands so their attention is focused on you. Make the commands extra fun and happy. End on a "Good Boy!" Then give your release command, "OK!" If your dog knows the Say Hello command, you can tell him to Say Hello to the new dog. If not, let both dogs wander over to each other. You need to stay put! Don't follow your dog. If you are too close to your dog you may trigger his protective instinct, you want your dog to be relaxed and meet and greet on doggie terms. Proper doggie greeting is side-by-side, each sniffing each other's backsides.
If you sense that the dogs do not like each other (hackles are raised, and neither one is demonstrating submissive behavior) both handlers should call their dogs to them in a happy voice and give them a food treat and praise when they get there.
What if your dog is known to be aggressive to some other dogs? You have two options.
1. Put your dog on a long lead. Give your dog the "space" he needs so that you do not trigger his protective instincts, do not allow the lead to tighten unless a fight is imminent and you are confident that you can pull the dog out of the fight area. If you allow the lead to tighten you could very well instigate a fight to break out. Consider using a harness instead of collar so you won't trigger your dog's survival instinct by placing force on his neck.
If a fight begins before you can react and call/pull your dog to you it is always a good idea to have a water hose handy. I know this is not usually feasible in a park! Some other tactics are throwing sand in the dog's mouths (so they will let go of their grip), or grass, or whatever ground you happen to be on. Sticks and other objects will also help them break their grip. It doesn't always work and I don't like to recommend it - but when you are in dire need something needs to be done. You can also lift up the aggressive dog's hind quarters. If it is a small dog, lift it up by the tail. A large dog may be able to turn and bite you (even your dog may bite you when he is in the middle of a fight), so I do not recommend this but I do like to let people know their options.
2. Put both dogs on a normal lead. Dog/Owner#1 on one side of the large area, and Dog/Owner#2 on the other side of the large area. Now, switch sides, circling a large enough half-circle to avoid the other dog, so that your dog stays comfortable. If you have a third person, have them walk around in the middle, always staying between the two dogs in a calm manner. If your dog is not comfortable, then the dogs are not far enough apart. 20, 30, or 50 feet are usually good places to start, sometimes you need to start farther apart. As you walk to the other side, talk to your dog, praise your dog, feed her food treats and keep her attention on you. Do this repeatedly, gradually lessening the distance between the two dogs. Dogs with real aggression problems need to gradually lessen the distance in days and weeks, not minutes or hours. Always end the session on a good, happy note.
If your dog lunges, growls, barks, etc. then you were too close, you may have tightened the lead, you may have gotten nervous. When this happens, walk (or run) directly *out* of the situation until you are far enough away for your dog's comfort level. Do some fun obedience with your dog and start again, further away from the other dog than you were previously.
Eventually, the dogs will begin to pass with only the distance of the third person (if there is one) between them. When they are comfortable with this, they may begin to pause and sniff each other for *1 or 2 seconds only!!!* They should be head-to-rear end, side to side, then continue walking away. Do this repeatedly. Only after your dog is consistently happy (not displaying aggression) doing this, can you allow the sniffing for longer periods. When sniffing occurs, keep the lead *relaxed* this is why long leads are important. The dogs will probably circle each other numerous times, and your lead may get slightly tangled, but it is important to keep it loose.
Once the dogs seem to have become friends, you can let them off lead to play.
Unless a dog is a known aggressor, all out fights are rare when two dogs meet on neutral territory. Observe their initial behavior and determine which dog is being dominant and which one submissive. Sometimes it is not clear. Two submissive dogs may simply run around and play. Two dominant dogs may do the same thing, but they may play rougher. As long as they are playing, it is a very good sign. But keep a watchful eye out for displays of aggression.
Most dogs thrive with canine companionship, however some dogs are used to being the only dog and do not wish to "visit" or "meet" anyone else for any length of time. If your dog is being "bothered" by the other dog and your dog constantly runs back to you as if you are "making" him meet this other dog, I would first, have him meet a more relaxed dog, and after that, reconsider your plans to add a new dog to the family. Some dogs are happier as the only dog in the household. Respect your dog and his individual personality.
If all goes well (and it usually does), play time can end with both dogs lapping up some water together, from separate bowls. Avoid food treats when both dogs are near as it could trigger protective instincts.
Next, simply put both dogs on lead and walk them home together!
When you get home, first let them off lead in your fenced yard. It will let the new dog explore the area and be less threatening to your original dog as he gets used to having the new dog around. When you bring them inside, you will have already prepared two separate eating areas, two doggie beds, etc. Let your dog, or the alpha dog pick which area he wants (even if its the new bed!) and read Two or More Dogs for more information on your extended furry family.