Yeah, I know, evasive title, but trying to squeeze the concept of "characters" into one post would jeopardize the future of this blog. I mean, eventually I'll run out of topics if I don't break the major elements of prose into small, self-sufficient chunks for my Thursday posts.
Anyway, characters of the protagonist variety should be strong in at least one of three ways: sympathetic, active, and engaging. The best protagonists are solid in two or all three ways.
A protagonist is "sympathetic" when he/she/it makes readers care about them. Having a strong voice and appealing tone help immensely in doing this. Everymen have an edge, as do frailer characters (although not 100% of the time). For the audience to really root for a protagonist the protagonist must be competent in at least one skill and succeed because of it. This is sort of a subset of "sympathetic." You might call it "rootablility." When you really respect and admire a character, you'll likely connect with them and experience sympathy when things go poorly for them. Protagonists with strong emotions and viewpoints tend to be the most sympathetic.
Howard Tayler and Dan Wells like to use the term "protag" as a verb to describe a protagonist being active. When the protagonist takes charge of the progression of the plot (woah, too much alliteration...), they keep the reader interested. Stories with active protagonists are usually tenser and have stronger conflicts than stories with reactive protagonists. In most stories, protagonists should be active more than 50% of the time (as a rule-of-thumb).
"Engaging" borders "sympathetic," but they don't blur a whole lot. Engaging characters draw in your interest, although they do not necessarily draw in your emotions. Superheros are typically engaging characters rather than sympathetic characters (at least on the surface). On the other side of the coin, seemingly incompetent characters can also be engaging because they often think in ways that differ from readers, which is often interesting. Often villains with terrible plans are engaging, yet totally unsympathetic. Characters that are sympathetic and active may evolve into engaging characters over the course of a story. It's up to the reader to determine what makes a character engaging, of course. I may be enthralled by a character you find drab. That's why writers try to write their characters to be interesting to their target audience.
A good portion of fiction today is character-driven, so it's important that your stories have strong protagonists if you want to sell them. While they don't need to be all three, your protagonists should more often than not possess at least two of the three characteristics of a great protagonist.