1. Educate yourself on their condition, and do so properly. Finding a few random articles that support your already-held beliefs does nothing. Ask them how you can learn, where you can find information. When you find it, ask them how it relates to their experience and condition. Just as no two cancers are the same, neither are two mental health conditions.
2. Talk to them with the intent of listening and learning, not the intent of proving yourself right or them wrong. Don't assume you understand their condition, how they feel, the reasons behind their actions or words. Nobody knows what their condition feels like except for them. You may be sympathetic, and if you've gone through something similar you may be empathetic. But you don't know how they feel or think because you aren't them.
3. Ask if they'd like you to attend a therapy session with them to learn more about their condition. The key here is to learn. The point of the session is for you to truly understand, to hear about their condition from a professional standpoint. It's not to talk about the things they do that affect you, nor to talk about your own issues. It's for your education and understanding. Not everyone will want this - don't push it if they don't. But to someone that does, going to their therapy session may well be the ultimate expression of love.
4. Love ALL of them. This doesn't mean put up with the cycling because you like "the rest of them". It means appreciate and value them as a person, and this includes their condition. It doesn't mean that you like when they feel this way, nor should you - they're in pain! Nor does it mean you like when it quite possibly gets taken it out on you. It means that you understand that without the emotions such as depression and frustration and irritability and anxiety, there wouldn't be the emotions like deep caring, concern, empathy, and love. If we don't have downs, we don't have the ups and vice versa. Without them, we're numb. I once read a quote that said (paraphrasing): Don't worry when I fight with you,worry when I stop. It means there's nothing left to fight for. Cyclers not displaying emotion, any emotion, is not a good sign. It means they have stopped feeling. It's a dangerous place. It may not just apply to their relationships. It may apply to their will to live.
5. We see the world differently. See the beauty in this. I'm much more likely to notice the beauty of a spring day, the wonder of sipping my coffee on my deck and watching my dogs play in the yard in the sun, than I am the laundry that needs to get folded. Logically, I know it needs doing. Emotionally, I can't pull myself away from the dogs and the spring day. This can, I'm sure, be frustrating a task-and-logic-oriented person who wants to get things done and feels like they're picking up the slack. Please understand that, as much as you physically and logically need us to do these things, we emotionally need you to stand there and enjoy the beauty of the sunny day with us. They're not right or wrong. They're just different. Perhaps work on reaching a compromise, or a way of understanding each other. Maybe, you could even learn from each other.
6. Love us the way WE need to be loved, not the way you do. This goes along with the point above. Not everyone feels, or expresses, love the same way. Expensive things do nothing for me. Emotional support does wonders. So while someone else might feel loved by receiving expensive gifts, I feel loved by having someone hold me when I cry and being sad that I hurt. It takes work, and effective communication, to learn how each other best feels loved.
7. Don't see the things you do for us as sacrificing. See them as loving. It's that simple. If you love someone, you want them to be able to be their true self (assuming it's not illegal or immoral), without feeling guilty or inferior. Of course, love is full of hard work and some compromise, because there are no two people who see every single thing eye to eye. But nobody wants to be a burden to the one they love. Don't make us feel that way.
8. Sometimes, you may just have to let go. I HATE to write this. I hate to say "maybe you should just leave a suffering person with a mood cycling disorder because you can't deal with it". But we all deserve someone who loves us for exactly who we are, not despite it. We all want happiness. And if you truly feel the two of you will just never be compatible, then perhaps, it's best to calmly, civilly part ways. No big fight and storm out, yelling how they're impossible and crazy. But truly, not everyone is right for each other. It's not fair to them, or you, to try to drag it on if you're not. It's only hurting you both, and the end result will be pain and resentment.