Every character walks a path that has two directions. While they move forward during the story (the character arc) it's important to remember to look back at the ground the character has already covered to get to that point. Everyone has a past, characters included. Their backstory or personal history gives readers a deeper sense that the character is complete, not just a stick figure going from one thing to the next.
Have you ever had those "ah-ha!" or "I didn't know that!" moments with your friends where you learn something about the way they were raised or something in their past? It's kind of like that. The character has a personality but his past is what has made him into the character he is at the time the story takes place. Another way to think of it is that the story being told doesn't begin at the prologue or the first chapter. For a more complex story it might begin years, even centuries prior. For most stories, the character's childhood is a good place to start.
Whether it's a terrible child hood, a traumatic event, a bad break up, it can be anything and it can make a difference in the way you (and your readers) view the character. When you give a character backstory, you create a life for them outside of the confines of the story being told. It makes it seem as if they exist independent of the story itself ... as if they are real, complex people and not flat fictional characters.
There are so many options to pull from but my personal favorite is exploring previous relationships.
If this is something you struggle with, you can start with the people that were closest to the character in his past or memorable life events. This usually means his parents, siblings, grandma, college roommates, former girlfriends, ex-wife, etc. Is the character still close to his parents? Does he get along with his father? His mother?
Many of my male characters are conflicted and struggle with maintaining a pretense of masculinity. I can give them effeminate reactions in the present story (screaming when he's scared, crying, failing to do something "manly" like change a tire, or successfully hunt deer), but I don't have to stop there. In order to give him a deeper sense of character, we can take it a step further and give him motivations tied to the present. Then, we can take it even further and create motivations for the present informed by experiences from the past? If you'd like, we could call this a dual motivation system. To take an example from my own works, Roger is effeminate because he's artsy in the present (motivation one) he's self conscious about it because his father was hypermasculine (motivation two). This instantly opens up a whole new world of possibilities as the reader is kept guessing which of these two motivations will win out as he faces a number of obstacles. It would be difficult to achieve this without backstory.
Suddenly, when the character steps out of his comfort zone and tries to defend his masculinity, we no longer run the risk having it feel like he is doing something out of character. If we know that he had a strained relationship with his father, we suddenly get to see a different side of the character... Yes, he's effeminate, but this suddenly gives depth to his struggles. He seeks male approval and it now appears that this has influenced his world view and planted the seeds of the insecurities he cultivates as an adult. And therein lies the genius of a multi-leveled motivational system (which is less complicated than it sounds).
Then there's romance... Was she a heart breaker? Or did she always find herself being dumped on a whim? Did she ever come close to marriage? Or was she actually married at one point? (Maybe she has an ex-husband off in Idaho or Prague.) Is she bitter and cynical because of it? Or is she still a hopeless romantic?
If the character is guarded and takes a long time to warm up, the reader will want to know why. The reason should probably be that she has been burned in the past (or maybe her mother was burned and she witnessed it). Childhood trauma is another route that many writers like to exploit but I personally feel that it can easily veer into becoming repetitive and predictable.
A character's backstory can clarify their motivations in the present, give them a more distinct personality, make them easier to relate to, and just make them all around more intriguing. It's fine to have a character that is a "strong independent woman" but we should ask ourselves why is she like that?
Not every detail of the backstory will end up in the final story (nor should it), but it will help you as a writer to know her motivations (who knows? It might come up later.) Of course, it's the writer's job to be judicious about what will remain but fleshing out a character's backstory is a great place to start if you're looking to develop deeper characters. So don't forget: it's not just about where the character is headed, it's important to also think about where they've already been.