How do you pray for someone? What specifically do you do or think about, and how long do you spend?

I think of them and I ask that God may bless them, enter into their lives or help them endure whatever is plaguing them. Not much to it really.

Do you get something out of praying; does it make you feel different in some way?

Same way speaking to a loved one, it's reassuring and comforting. At least, some of the time. Some times prayer can be desperate, and you can end up begging and crying.

Do you pray at any specific times or locations, or just do it whenever the mood strikes you?

For me at least, I tend to pray right before I go asleep, when I wake up, and I try to pray before each and every meal and give thanks. Sometimes I'll pray when something is concerning or when I'm under temptation.

If you're the target of someone's prayer, do you know it, and when do you know it? (at the moment of prayer, or at some later time?)

Don't think I've ever felt like someone was praying for me. I know people have but I've never "felt it", so to speak.