1. Consider the relationship. Just because a man may notice something about me does not make it appropriate for him to comment on it.
2. With non-relatives, be very cautious about giving complements you would never give to a man. You would never say something like, “You’re such a sweet little man and you always dress in the cutest clothes!” You might say, “I appreciate how kindly you treat others. And I like that jacket. It looks great.”
3. Treat the object of your compliment as a person, first and foremost.  You don’t need to keep reminding her that she is a woman, or act as if you can’t get past her femaleness. (See #2. The way I adapt these guideline is by not complementing a non-relative male on any masculine attribute — which would most likely be inappropriate anyway — and by not adding the slightest hint of a sexual overtone by emphasizing his gender. I might say, “You have a unique musical talent,” but I won’t say, “You are such a talented man.”)
4. Don’t use “compliments” as a means of embarrassing women and girls. Be succinct and make sure the compliment is welcome. Compliment something you honestly believe the recipient would want you to affirm. If not, apologize (“I’m sorry if I was out of line” or “I’m sorry for embarrassing you”) and then be quiet. 
5. Don’t be effusive and don’t gush. That’s just creepy.
6. Make sure there is an altruistic purpose to your compliments. As Christians, we should encourage one another, but we need to be careful that we are encouraging the right things in the right ways. The compliment should be for her benefit, not yours. Don’t use flattery to manipulate. Try to never put a woman in the awkward place of wondering about your interest or motives, or being creeped out by your attention.
7. Be specific. “You’re terrific” or “you’re nice” can sound like insincere flattery or an attempt at flirting. If sincere, it communicates nothing of substance, other than “I like you” — and such declarations are usually, depending on the circumstances, unnecessary or best left unsaid. (See #1) If the compliment is to be meaningful and beneficial to the receiver (see #6) then it needs to be specific: “Your tireless work behind the scenes does not go unnoticed or unappreciated”; “I admire how you stood up to that guy — that took courage and grace”; “You have a way of making us all feel welcome, as if we are your personal guests”; “You are so patient with our son, and so encouraging”; “You have a way with words and a level of insight I appreciate — your latest article is just one example”; “You made the unpleasant ordeal of picking out a new washing machine not an ordeal at all! You were so good at sizing up our needs and wants, and helping us find the right machine. You made it easy, so easy it was almost fun! You have a great sense of humor — I had no idea laundry could be so hilarious.”
8. Compliment only what you have observed to be true. This can include repeating a compliment you’ve heard/read, as in, “Wow, your play sure got rave reviews!” or “Your parents told me you did great!” But don’t offer meaningless speculation or assume anything about her character, experiences, beliefs, future plans, etc. You might end up looking like a fool or — far worse — might inadvertently make her feel terrible.
9. Keep your compliments appropriate to the context. If it’s a professional setting, be professional. Affirm her professionalism, work, knowledge, skill, etc. but don’t intrude on her personal life or pretend to “know” other things about her. (See #7. I don’t know any woman who, in her professional life, wants some relative stranger to insist, “I can tell you’re the sort of person who…” especially if he them spouts meaningless flattery.)
10. Unless she is about 6 years old or younger, avoid words like “cute” or “adorable” — unless you are describing her kitten. Condescending “compliments” are insulting. 
11. Try to understand that not every woman has a need or even desire to be affirmed by random men. I know that men are often hurt when women — even complete strangers — don’t “appreciate” their compliments. But, except for the most insecure and desperate among us, we don’t need male validation from every Tom, Dick and Harry as much as some men seem to think we do. Having a guy who barely knows me pronounce his positive judgment on me doesn’t necessarily feel like a “compliment” as much as intrusive and nervy. His opinion of me doesn’t matter as much to me as it does to him; often I’d rather he have kept it to himself. So don’t assume women and girls will respond with grateful enthusiasm to your compliments, and try to understand if they act annoyed and put upon. Most of us have already had to endure too much of men judging us and it’s hard to be gracious about it. Plus, there can be a lot of anxiety there too, for good reason — a seemingly “harmless” compliment can suddenly go terribly, terribly bad. However, well-mannered, meaningful, succinct, specific, and appropriate compliments tend to be welcome by almost everyone, male or female.
12. Avoid the “back-handed” compliment. This means you shouldn’t include any insult or sign of disrespect, no matter how cleverly worded, into your compliment. “You’re not like other girls” is not a compliment, nor is, “You think like a man”.
13. If you mess up, apologize and drop it. Never berate a woman for not responding as you had hoped. The words “Can’t you take a compliment?” should never cross your lips.

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Edited to add this:

14. Or maybe I should call this 1A. Consider the relationship before you address people. I realize that there are cultural differences regarding when terms of endearment are appropriate or not. But, it’s a safe bet that, if we’ve just met and have never been formally introduced, that it’s a bit premature in our “relationship” for us to be considering each other as “dear”, “honey”, and “sweetheart”. Perhaps we should reserve those “sweet nothings” until we get to know each other a bit better. Unless, of course, you’re just being an absent-minded mom of many kids, who doesn’t realize that she now calls everyone — her kids, her husband, the dog, the cat, the mailman, and her pastor — “sweetie” or “honey”…either that or something that sounds like, “Math-Mi-I-Ben-Dan-Jesse…oh, never mind!”