Relationships in Batman's world are somewhat fraught and tense, mostly due to the nature of Bruce Wayne/Batman himself. He's not the warmest person, and his paranoid and secretive nature nearly ensures his relationships will be at arm's length.
There are, however some people whom Batman has let into his tight inner circle, people who have earned his trust as he has earned theirs. They're the ones to whom he knows he can turn when he's in trouble and who can turn to him when they need help. These are the relationships Batman will actually work on so they don't fray too much.
Ace The Bathound And Bruce Wayne
Dogs are man's best friend, and that holds true even for aloof, billionaire superheroes like Bruce Wayne. A dog will love unconditionally, and Ace, in one form or another, has been with Bruce for a long time. He's one figure in Batman's life who, no matter what, is always happy to see him.
Ace is a stabilizing force in the life of Bruce Wayne, as much as is Alfred but in a different way. Ace doesn't judge Bruce or his choices. He's just there to offer love and affection in return for some belly scratches.
Leslie Thompkins And Bruce Wayne
Leslie Thompkins runs a clinic in an area of Gotham that used to be known as Park Row, one of the tonier parts of Old Gotham. It was where Bruce watched his parents get gunned down in front of him. Leslie was the person who found the frightened little boy and comforted him when probably no one else could have. She never forgot that little boy, and he never forgot her.
She runs her free clinic in what is now known as Crime Alley and helps all who walk through the door. She knows who Bruce is, and Batman has come to her in the past with medical needs. As Bruce Wayne, he helped fund the clinic.
Kathrine Kane And Bruce Wayne
Bruce Wayne has few surviving relatives, but comics fans know Kate Kane, aka Batwoman, is one of the few. Bruce's mother was Martha Kane, and Kathrine is her niece. Presumedly, she and Bruce have known each other since childhood. They have both been through parental tragedies in their lives, witnessing the deaths of parents and having to deal with the grief.
Kate was from a long line of soldiers, and her ways of dealing with criminals are at odds with Bruce's. Kate has shown no hesitancy about the use of guns while Bruce continues to eschew them. Batman has trusted her with missions and asked her for help occasionally, even offering her a place in Batman Inc., which she refused.
Dick Grayson And Barbara Gordon
They have both been through a lot, Dick and Barbara, and probably know more than anyone what it's like to be under the mentorship of a man like Batman. They have been friends, lovers, even engaged. They may occasionally part for a time, but they always come back to each other.
Dick and Barbara have a long and complicated history, but both have been there for each other when needed. Perhaps most significantly, they both are part of the Bat family, sharing a rare understanding about what that means. Each knows when the other might need help, and they've been together long enough to know when to offer it and when to leave it alone.
Batman And Superman
Batman and Superman have clashed in the past, but both of these founding icons of DC Comics have a strong moral code and a belief in justice. They don't always agree on the application or the degree of that justice, but in the end, they agree that evil must be stopped. They have a mutual faith in each other and consider themselves friends.
Still, just because they have faith in each other doesn't mean they have total trust. Superman is probably more trusting of Batman than Bats is of Supes. That's why Batman still has that chunk of Kryptonite, just in case Superman oversteps and loses sight of his moral code. It's that paranoia that keeps them from being closer, but if Batman needs help, Superman will be there.
Alfred Pennyworth And Dick Grayson
More than even Batman and Alfred, Alfred and Dick are friends and at times allies against Bruce in his more obstinate and self-destructive moods. When Dick was growing up in Wayne Manor, Alfred was the warmer parental figure. He often took Dick's side when being a kid was more important for Dick than solving the latest puzzle from the Riddler.
Today, Alfred is the one person Dick will go to when trying to gauge what Batman's thinking or what he might be up to, as he's not up to speed on the day-to-day operations. The two men will often try to talk Batman into a different approach to something, and both act as sounding boards to each other and to Bruce — when Bruce lets anyone in on his thinking, that is.
Batman And Commissioner Gordon
Besides Alfred, Jim Gordon was the Batman's first real ally and one who has fought for and stood up for Batman against those on the police force or the political machine that would try to take him down. It's not a one-way relationship either, as Batman has fought for Gordon, helping him navigate the Gotham political world as billionaire Bruce Wayne, using his pull to protect Gordon.
There is a deep trust between Gordon and Batman, one that keeps the relationship going through tough times and when Batman's secrecy becomes maddening. Gordon knows that Batman knows what he's doing, so he helps him do it. The relationship has been tested, and Gordon has had to reign in his friend from time to time, but the friendship is too valuable for either man to let it falter.
Batman And Nightwing/Robin
Dick Grayson has always been Batman's best partner, and the one he still works the best with on missions. Dick is able to follow Bruce's thinking and his plays because Bruce trained him that way when Dick was Robin. Since leaving and branching out as Nightwing, he's become his own man and brings his experiences back to the team when he and Batman reunite to help each other out.
Dick Grayson and Batman met on the worst night of Dick's life, and the two orphans share the pain of witnessing the deaths of their parents. There is a deep trust between the two men. Dick is Batman's most trusted ally. Bruce even let Dick put on the cowl to protect Gotham when Bruce was unable to. Dick has also been a mentor to other Robins, particularly Tim Drake. Even after arguments and clashes, especially over Bruce's secrecy, the two men still love and respect each other. Batman has even stated that Dick is probably the better man.
Batman And Alfred Pennyworth
Whether Bruce admits it or not, Alfred is Batman's best friend and the closest relationship he has, even closer to Bruce than his own son. Alfred helped raise Bruce after the deaths of his parents and has kept a parental eye on him ever since, sometimes reluctantly going along with Bruce's mission as Batman.
Alfred has been there for every step of Bruce's journey and has provided aid for every one of those steps. As Bruce's protector, he has kept all the Wayne secrets and those of Batman as well. It can be a heavy burden, watching a loved one put themselves in danger night after night, but Alfred bears it well. He is one of the few who see through the mask to the man behind it and is able to read Bruce's emotions and needs. It's doubtful Bruce would be able to function without Alfred backing him up and keeping him going.