Candice - You appear to me to be a loving person who in your dream is exploring and experiencing the different forms of love.
You start with the fundamental love a creature has for it young, and it transfers to you by settling on your hand. The puppies love - puppy love - which has a lot of awkward playfulness in it - and the white horse that is beautiful and can carry you about is the sort of love that has been transformed into giving outside of its natural instincts.
You are yourself the end product of many different animals, and so still carry them in you, and your dream - an unusual one - indicates you love and are loved by these parts of your own nature which puts you in empathetic contact with external animals. Or perhaps it is the other way around - your love for animals has enabled you to be loved by or love the animals within you.
Please see
http://dreamhawk.com/dream-encyclopedia/brain-levels-and-dreams/Tony