I hate picking up a book or going to see a movie only to feel like I have just met a set of cardboard one-dimensional characters that didn't make me feel anything. It feels like a waste of time where I could have done something productive instead. One of the worst things for a writer is to create characters that readers do not want to get emotionally invested in, nor commit to keep reading about them.
When I sit down to write, making my characters multidimensional is something I have to be constantly conscious of, and work at getting it right. It is not an easy task.
So what makes a character multidimensional?
Well, think about all the people you know in real life. What makes them interesting/different/human? You would be hard pressed to come up with one person in your life who always has the same expression and acts the same way no matter what the circumstances. Everyone has their own little neurosis, occasional moods swings, periods of depression or anger following a negative event and unbridled joy following life's positive events. Everyone dreams of something, has certain hardships in their daily lives and loves in their own way.
A character who always acts confident no matter what's going on around her may become flat and boring. Make her act confident in public but give her moments of insecurity and doubt in private, and suddenly she becomes more human, and definitely more relate-able. Hey, even super-heroes show weakness from time to time. That's why they are interesting.
Observe the behavior of people that you interact with every day (or on occasion) and note the nuances of their speech, body language and mannerisms. Write these down and keep adding to the list frequently. Then use these notes to make your characters come alive. Infuse all your characters - primary and secondary, protagonists and antagonists - with the traits and behaviors of real, "recognizable" people.
Another way to make characters more multidimensional is to give them a history, a background. It is always easier to understand individual choices and personalities when you know what shaped them in the past and what they want in the future. None of us exist in a vacuum - don't insert your characters into one.
Have fun with writing your characters, explore them and let them tell you about who and what they are as you write them. Allow them to giggle at an inopportune moment because they are nervous, and to have a good snotty cry when life gets too tough. Make them come alive for yourself first. Then they'll come alive for your readers as well.