If evolution is real, then why are there still apes and monkeys on Earth?

There are some common misconceptions regarding evolution that I think may have prompted your question. The first misconception is that humans evolved "from" apes or monkeys. The truth is that humans are a kind of ape, although a relatively hairless variety, and we are essentially evolutionary "cousins" to other apes rather than the evolutionary "grandchildren." All existing species of apes descended from a common ancestor, and before that there was a common ancestor which gave way to monkeys and apes. 

The second misconception I believe to be at work here is the common opinion that humans are the ideal form. Some people feel that humans are the best species to ever have existed, and that some day, all existing species will give way to human-like forms. I will explain more in just a moment about how this is not even possible, but for now I would like to mention that it is teleological. Teleology assumes that all changes are in a certain direction, with a predictable or desired end result. In this case, Homo sapiens or being human-like is the desired, valuable end result. In reality, being human-like is only valuable to us--humans! A fish has no desire or need to be human-like beyond basic characteristics that all animals share. Fish don't have a need to walk on two legs or communicate socially. If they did, they wouldn't be fish.

Evolution is the result of two major forces--one is mutation (lots and lots of random changes in genes) and the other is environment. When a certain gene or set of genes helps an organism to thrive in their environment, it is more likely to be passed on. Environments are constantly changing, though, so organisms have to change to keep up! Evolution describes this long-term process of changes which help an organism to survive in their specific environment. So, returning to your original question about why monkeys and apes still exist, it's because these species are well adapted for their environments, and we humans are well adapted for ours. 

The Smithsonian has a great website all about human evolution and where we came from! There's even a "family tree" depicting the many evolutionary changes that have occurred throughout our history. If you look at some of the photos of faces and bodies of the many species we can consider "cousins," you may see that as you go further back in time, the more species in our lineage resemble chimpanzees. That's because humans and chimpanzees are descended from a common ancestor. If chimps are our "cousins," then this (as yet unidentified) common ancestor is "grandma"!