Great question- one I'm sure I'd have if I was a singlet. Also though, one that is difficult to answer. I'll give it my best shot- but please understand, I do not pretend to speak for everyone (or anyone!) who has DID, as we all experience this disorder in VERY different ways. This is just what it feels like to me.
To start off, I want to clarify that I understand the science behind DID. I do actually recognize that I am fully one person. I get that saying otherwise would make me look a little kooky- and that's not what I'm doing at all! I'm going to try to explain how it FEELS- not how it actually is. So bear with me!
I, myself, in my experience of day-to-day life, am me. Just me. Much the same, I would guess, as you are you. I'm never anyone else. I'm never a different gender, age, etc. My name always stays the same. I always believe the same things, and have the same values. I'm just me. That's how I feel.
I have a body. It is mine, but it is not me. Its just a body. Empty, useless, if not for my occupation of it. It serves me well, I feed it and take care of it, but its just a body. I live in it. That's how I feel.
Inside the body, there are more than just me. (that's how I feel) Some of the others in here, I know, some I have not met. Some of the ones that I know I like, others I do not like. Some believe similar things as I do, others do not. Of those that do not, I respect the views of some, and I think others are batshit crazy for the stuff they think. But they are not me. They are them. They have names, genders, ages, etc. Some of these things line up with my identity, but some do not. It doesn't matter, in much the same way as if I was standing in a crowd. Some people in the crowd might also be female, or also be my same age, or also share my same first or last name. It wouldn't matter- they are them, and I am me. That's how I feel.
As I look around in the world, I see that most people get one body, and one self. So they are always there. They are always the one using their bodies. When they fall asleep, they are also the one that wakes up. They participate in a wide range of activities, from playing playdough with their kids, to watching horror movies, to having sex, to doing taxes. Its all them. They are good at some things, and not so good at other things. But they are the ones that do all the different things that make up their lives.
Not so with me! (at least, that's how it feels!) Sometimes I'm in my body, like you, and sometimes, I'm not. When I'm not, there's nothing. No time passes. No dark, no light, I'm not in some kind of suspended animation....its more like I pushed the "skip" button on the DVD player. Sometimes, it works out, like if you were watching an old DVD that you knew well, and you skipped a boring or scary part. Sometimes, you accidentally skip your favorite part. And sometimes, its as if you were watching a DVD you'd never seen before, and you missed key parts in the story. You have no idea what's going on, who's doing what, etc.
It can be really scary, heartbreaking, and painful- because during the time I am gone, someone else is using the body. Sometimes, they are doing things I would do, or would like them to do, like going to work, or fixing my computer, or exercising. Other times, they are doing things that are important to them, but are abhorant to me. In some ways, its their choice, their life- but in other ways, it sure as hell is not. My job in therapy then, is to work with my parts, so we come to an understanding about the way "we" (which is all one brain, remember) live this life. Its kind of a crazy process.
As I process the feelings, memories and skills that each part holds, we become closer and closer until we merge completely into one. Ultimately, all of us will have one single stream of consciousness- this is commonly called "integration" of parts. As this process happens, the feeling of "switching" changes as we become "co-conscious". First (the worst part!) I start to be able to see what's going on while the part is using the body. Its like watching your life as a first person movie. No control, no choice, just watching. Sometimes, this is terrible. But ultimately, as I continue to work, co-consciousness begins to change for me/us. Instead of one front and one stuck watching, its like we are making choices together. A team, but still two. Then, finally, the two become one. Integration. Neither have to loose time, neither have to watch choices be made they hate, because there are no two. No red playdough, no blue playdough, but a better, stronger, bigger ball of purple playdough. That's what it feels like to me.