So how important is sweating when working out? And what does it mean if you work out hard, like you are breathing heavy and can't talk or sing but you are barely or not at all sweating?

I think some people are just different. I run with my mother and I sweat a lot and she barely breaks a sweat. We are both working hard, but she has just never been much of a sweater.

Sweat does not indicate exertion. It is a cooling mechanism for the body. Some people sweat when they are nervous and that isn't a workout.

Being out of breathe during exercise is normal but NOT if you are gasping for air. That may be more serious and you can tell the difference between normal heavy breathing and abnormal breathing. The more in shape you become your breathing will mellow out. When you work out, it also works out the cardiovascular and respitory system. Usually the healthier you become the more in control of your breathing you become. Now when I work out I can't sing the Star Spangled Banner but I can have brief conversations, give short answers, count reps that sort of thing.

Indeed, you should try to keep your tempo at a point where you can still talk a bit while going, like RoyalAthena said. Personally, it makes it easier for me to keep going when I'm going at a pace like that.
I always sweat a bunch, and I like the feeling of having worked hard enough to produce sweat, but realistically it doesn't mean anything.

I could give short answers, in fact I do because I have two little kids that love to work out with me. lol But I couldn't carry on a conversation or anything. I was doing the 30 day shred and that's what keeps me from being able to talk a lot and breathing heavily, but I am definitely not gasping for air. Thanks everyone for the help!

I have heard from many sources that sweat isn't that wonderful of an indicator for how hard you are working out. It's simply the body's cooling mechanism. I would judge the intesity of your workouts based on how you are feeling, check your HR and find out if you are in your target HR range or possibly over or under your range. I take my HR manually, I count the beats for 10 seconds then multiple by 6 and I have my HR. I like to do that 4 or more times during my workouts to make sure I am getting the best workout I can without overdoing it.