I will give you my example. I used to be mortally afraid of dogs - or any other fairly-sized animal for that matter. There is an ulterior reason to this which I am still trying to unravel. Anyway, my fear of dogs was so paramount that I would simply refuse to visit my relatives or friends or neighbours who had a pet dog.
Then a couple of years ago, a family moved right next door to us. They have a she-lab for a pet. The lab, Ginger, is so loved and pampered that she basically does not fear anybody. And labs generally have heaps and heaps of energy. This energy coupled with the general fearlessness, was not something I could digest very well. I was ****-scared of Ginger and I just plain refused to step into the same elevator as her once. I feared being confined in limited space with something that had a booming bark and extremely sharp teeth. Thankfully, neither the family nor the dog took offense to my avoidance.
Once when I was coming home from work and just getting out of the elevator, Ginger cornered me. She was basically waiting for her Dad (my neighbours treated her like a daughter rather than a dog!) to come home from work. At the sound of the elevator, she came running expecting to see him. But when I came out, she didn't know what to do. Her first instinct was to jump up and put her front paws on my chest and lick my face fervently (which is what she does with everyone, even my family). But she also knew that I was out-of-bounds. She was told she couldn't go too near me, because my neighbours knew my fears. So she just stood there between me and my house gate, looking up at me with the saddest-cuttest eyes ever. I passed through keeping my fears under control, but by the time I reached my house gate, she had won me over. I called out to her and she was all over me in a second.
That's how I got over my fear of dogs. That is by far the simplest way to do that. Confront your fear and let a gentle soul bowl you over. Dogs are lovely creatures, they love people and love to be loved just as much. If you get to know a dog well enough, your fears will vanish soon.
Ginger and her family moved away last year, but I visit her every now and then. She still jumps up on me, but knows that licking (or kissing as she would call it if she could talk) is not an option. :)