Located in Ubud, the Sacred Monkey Forest is a great place if you want to see the local macaques and take in some of the temples.
It's a sacred site for Balinese Hinduism, the religion is a lot different then Hinduism practiced in other parts. It combines aspects of Animism, Ancestor Worship, Buddhism, and Hinduism. Most villages have 3 main temples each dedicated to different gods and purposes. The temple located in the monkey forest is the villages Pura Dalem, associated with the God Siwa, who is known for judging a person's karma and preparing them for reincarnation. Even if you're not all that excited about monkeys, it's worth checking out this forest as the temples and statues are quiet interesting.
As for the monkeys, while cute and usually pretty mellow, there are a few safety precautions you should take. Don't bring a lot of belongings or food into the park. Personally I recommend just leaving everything in your room, Ubud is small, and you can easily run back later and get things. Just bring enough for the entrance fee (about 30,000 rupiah a person) and your camera. Most problems caused with the monkeys are due to food or belongings that they got into. They're wild animals, and curious ones at that, they aren't going to be on their best behavior around temptations.
So leave all your worries and belongings behind, and take a stroll for a few hours amongst the monkeys, the trees, and the deities. You won't regret it.