This is a topic that I've had a lot of discussions about. And they all come down to a few things:
1. Are they believable?
2. Are they compelling?
3. Are they individuals you can identify with?
4. Do they work for the story?
This is probably the most subjective element in a character. Personally, most of the beta-male [weak and indecisive] characters are not believable to me. Most men I've known in my life are anything but this.
So, when I write a character I make them over-the-top, no holds barred alpha males that will do whatever it takes to accomplish their goals -- no matter how dark or twisted these goals are.
Some would argue most humans, regardless of sex, are weak and indecisive and it's interesting to read about how they manage or do not manage their lives in this respect. I'm just not one of those people. Normally I will put down a book like that, because I would not find this believable or worth the effort and/or time.
This is not to suggest that the alpha male characters are right, or wrong [moral questions are worthless when dealing with people's goals and drives -- morality is what we think people should do...not what they do], but that they are more interesting as villains, heroes, or tragic figures than an Oblamov character is [see Ivan Goncharov's 'Oblamov' for more information].
This should flow almost directly from 'Believable'. There are important elements here, however, which distinguish it from believability. As an adjective it literally means: tending to persuade by forcefulness of argument. This is the key, the character must have a point of view and be willing to pursue that perspective through to the, perhaps, bitter end. Actions in such a character need to be bold, shocking, controversial, and unwavering -- even if the character has doubts about their actions. Actually, it's a good idea to have your character confront their doubts, but still to forge ahead: Shakespeare and Homer have taught us that if naught else.
However, endless and nauseating self-reflection may make for beautiful prose but will not help the story or plot [the distinction between these two is not for this post] in the end.
Identifying with the Character
Here is a psychologically sketchy area. Some characters with all the appropriate virtues and mild vices are easy enough to identify with, but often this can be boring. Sometimes it is fun to take a ride in a character we might find repellent in our actually lives, but in our dark hearts [and everyone has one] they can be interesting, enlightening, and even entertaining.
My writing is concerned, almost exclusively, with the Anti-hero [An anti-hero is only good in comparison to those around them], and therefore identifying with them is a problematic enterprise and one that is possible only if you carry a seed of this darkness in yourself. Thankfully, most of us do.
Does the character work for the story?
An anti-hero in a bodice-ripper can work, but it isn't the norm. Now I am speaking of the genuine anti-hero and not the milksop Byronic wannabes of Twilight, Anne Rice, or countless other creations of the Chick Lit. World [my prejudices are here for the world to see].
An anti-hero can work best in a world of moral ambiguities or outright rejection of moral and legal systems which bind humanity one to the other in a social and legal construct of acceptable and unacceptable behaviours.
Ultimately, a character is a manifestation of a book's theme and aesthetic sensibility and must work and act within that zeitgeist. If the reader cannot accept the world view then their ability to be convinced by the character will come into question and the book, in the end, will be a failure for the reader. It is for this reason that we now have so many genres, sub-genres, and such a plurality of general fiction and literary fiction today. We are literally drowning in literary micro-niches [dig through Amazon some day if you doubt me].
As a final thought, let me suggest this list of elements for a successfully character is not complete -- how could it ever be? What defines a character is a nebulous area, as is what defines plot [a good one at least]. However, these are the elements, for this writer, which helps to define characters that are worth my time and, hopefully, yours.