I'm a writer and thus am always seeking to be inspired to write stories out in ways where I can be truly happy with them. The ideas will already be in my head, however, finding a means in which I can get my focus to truly express those ideas (and in a way that flows and is also logical as well as 'good'), is truly important. That said, what is inspiration to me so that I can achieve this? Well, it's an emotional feeling, one that allows me to find that focus to express the already formed ideas.
I have found that my triggers for inspiration are quite varied. Sometimes it can be something obvious like music or a scene from a movie. Other times it can be more abstract, like when I just spend time with my kids and an emotion hits me the right way. The emotion, once triggered, might not even be a result of something that seems obvious for what I'm trying to write either. In other words, it really can depend on my current mood. One thing I will say, however, is that when I seek out inspiration with a certain level of gusto and force... it has a tendency to elude me and make me more frustrated. As such, I try my best to let go and let it find me instead. Not always easy to do either. >.<
Also, once inspired, I tend to almost get tunnel vision. It's as if no one can break me from my zone. I will just write and write, holding onto that emotion for as long as it will allow. It's in those moments that I tend to be happiest with my writing (even if it still needs to go through the wringer with editing and proofing and other such things... that can come later, getting the initial idea and story out is something else).
As for why there can be difficulties in becoming inspired? I think, for myself at least, it has to do with either stress or distractions. When stressed, it's hard for me to focus on anything, even the simplest of things. My mind becomes a whirl of crazy and thus has no desire to even seek focus until the stress levels drop some. This then leads to the idea of distraction. When stressed, distraction comes easily. It can allow us to cope and push through those times. But, there are also times where distraction just comes more easily and thus, if you hit a bit of a stumbling block, it can be embraced without much effort. I think the key here when such things occur is to use the distractions for your benefit. I know I've been trying to use such times to work on thought ideas or characters or as a time to do research on things that might be needed down the road for a story I write. So, basically, I try to make my distractions as productive as possible. Of course, this has its moments of success as well as a lack thereof.
Anyway, that's my long winded version of inspiration and what it means for me and how I cope with seeking it or letting it find me.
If anything else pops into my mind or if someone else posts something I agree with or have experience with, I'll add more then.