I'll add some thoughts to this.
First off, an engaging character is not necessarily one we sympathize with. Engaging and sympathetic are not mutually exclusive, nor are they opposites.
I think an engaging character is one who captures the reader's interest. The reader wants to know more about them and hear more of their story. This doesn't mean that the reader likes the character, however, and it does not mean they necessarily agree with the things they are doing.
Antiheros can engage us even though they may be performing actions that are abhorrent or detestable. Walter White from Breaking Bad comes to mind. He becomes less and less sympathetic as the series progresses, yet he remains an engaging character throughout.
So what makes a character engaging, then?
I'd like to share a quick image:
This comes from a study in creating characters by Aaron Ehasz, head writer of Avatar the Last Airbender. I think it does a great job showing what qualities a character must possess to be engaging.
One way characters become engaging is when we see into their heads. We see the logic behind the things that they think and the decisions that they make. Yet we may, at the same time, not agree with their thinking. Characters can be engaging when they think differently than us.
An engaging character is also usually a deep one.
We don't need to know what their mother ate for breakfast on the morning he/she was born, but showing enough backstory and depth can go a long way to making a character compelling. They should have multiple dimensions and be able to think about things in more than one way. Good internal conflict comes from a character struggling between two ways of thinking or two ideologies that are equally important to them. It shows that they've experienced many things and can think about the world from multiple angles.
Here's something to consider: who do you like to hang out with in real life? Why are your friends engaging to you? This can highlight some attributes that you might find engaging in a fictional character.
Now what about pitfalls?
Marry Sues have been mentioned (people who are overly idealized or perfect). I think another pitfall would be projecting engaging attributes onto side characters, to the neglect of the main character. A lot of writers tend to create quirks or interesting qualities for side kicks or friends of the protagonist.
What happens, though, is that readers get left with a blank-slate main character who is surrounded by people who are much more interesting and colorful than they are. This is how villains sometimes steal the spotlight. They become more interesting than the heroes are.
Why would a villain be more engaging than a hero?
Part of it, I think, comes from predictability. Villains are the ones who have to come up with clever ways to thwart the hero, while all the hero does is react. An engaging character can be someone who simply does the unexpected.
What's superman going to do if he sees a plane falling out of the sky?
He's going to catch it.
Just like he catches every plane.
That's BORING.
A sense of the unexpected can really make a character interesting. You want them to act in ways that are believable, of course, and there need to be reasons behind their actions. But a character gains depth when they act in a way contrary to expectations. It tantalizes you, making you want to know why they did what they did and what they were thinking.
Never forget the all important human error, too. People make mistakes. 
The reason I posed the last bonus question was mostly to have people think about what a story would be like without an engaging character. It is a silly question, perhaps, but one that's worth asking.
Should you set out specifically to create an engaging character? Or should you be setting out to write a story? For some, characters being engaging is just something that grows out of the process of writing. Not all writers tackle character creation with a checklist. There's no magic criteria that will ensure your character comes across as engaging to all readers.
At the very least, your characters should be engaging to you, If they aren't engaging to you, they sure won't be for your readers. So make them interesting!
I'll add some thoughts to this.
First off, an engaging character is not necessarily one we sympathize with. Engaging and sympathetic are not mutually exclusive, nor are they opposites.
I think an engaging character is one who captures the reader's interest. The reader wants to know more about them and hear more of their story. This doesn't mean that the reader likes the character, however, and it does not mean they necessarily agree with the things they are doing.
Antiheros can engage us even though they may be performing actions that are abhorrent or detestable. Walter White from Breaking Bad comes to mind. He becomes less and less sympathetic as the series progresses, yet he remains an engaging character throughout.
So what makes a character engaging, then?
I'd like to share a quick image:
This comes from a study in creating characters by Aaron Ehasz, head writer of Avatar the Last Airbender. I think it does a great job showing what qualities a character must possess to be engaging.
One way characters become engaging is when we see into their heads. We see the logic behind the things that they think and the decisions that they make. Yet we may, at the same time, not agree with their thinking. Characters can be engaging when they think differently than us.
An engaging character is also usually a deep one.
We don't need to know what their mother ate for breakfast on the morning he/she was born, but showing enough backstory and depth can go a long way to making a character compelling. They should have multiple dimensions and be able to think about things in more than one way. Good internal conflict comes from a character struggling between two ways of thinking or two ideologies that are equally important to them. It shows that they've experienced many things and can think about the world from multiple angles.
Here's something to consider: who do you like to hang out with in real life? Why are your friends engaging to you? This can highlight some attributes that you might find engaging in a fictional character.
Now what about pitfalls?
Marry Sues have been mentioned (people who are overly idealized or perfect). I think another pitfall would be projecting engaging attributes onto side characters, to the neglect of the main character. A lot of writers tend to create quirks or interesting qualities for side kicks or friends of the protagonist.
What happens, though, is that readers get left with a blank-slate main character who is surrounded by people who are much more interesting and colorful than they are. This is how villains sometimes steal the spotlight. They become more interesting than the heroes are.
Why would a villain be more engaging than a hero?
Part of it, I think, comes from predictability. Villains are the ones who have to come up with clever ways to thwart the hero, while all the hero does is react. An engaging character can be someone who simply does the unexpected.
What's superman going to do if he sees a plane falling out of the sky?
He's going to catch it.
Just like he catches every plane.
That's BORING.
A sense of the unexpected can really make a character interesting. You want them to act in ways that are believable, of course, and there need to be reasons behind their actions. But a character gains depth when they act in a way contrary to expectations. It tantalizes you, making you want to know why they did what they did and what they were thinking.
Never forget the all important human error, too. People make mistakes.
The reason I posed the last bonus question was mostly to have people think about what a story would be like without an engaging character. It is a silly question, perhaps, but one that's worth asking.
Should you set out specifically to create an engaging character? Or should you be setting out to write a story? For some, characters being engaging is just something that grows out of the process of writing. Not all writers tackle character creation with a checklist. There's no magic criteria that will ensure your character comes across as engaging to all readers.
At the very least, your characters should be engaging to you, If they aren't engaging to you, they sure won't be for your readers. So make them interesting!
Still redundant. If you know what an engaging character is, you will know how to write one. You could ask Is there subtext with the character? 
