The stickied Fitness Forum faq has a number of exercise programs recommended including some that don't require any equipment and are great for starting out.
Just remember moving is the key for cardio, so you can just do things like take your kids to the park and chase them around. And for resistance training, you don't need to go to the gym or own a bunch of weights, you can use a backpack full of books, or even lift your kids to work the muscles.
As for how long you do them, if you're just starting out try to get 30 minutes (or less if you're really out of shape) in 4-5 days a week to start out and build up to what you're comfortable with.
Just as karozel says, the FAQ has a great deal of good information in it that you might find helpful.
If you want to stay in the house, for cardio you might consider a stationary bike, elliptical, or treadmill. For strength training a basic set of light dumbbells/free weights and a few simple exercises can get you started.
Good old-fashioned calisthenics would also be a good path to follow: push-ups, crunches, lunges, squats, jumping-jacks, ect. It might seem weird to be doing all these exercises they made us do in middle-school PE, but they are excellent for your health and fitness, particularly if you are just beginning. Best part... they are free and you can do them in the living room.
A good number to shoot for is to work-out for at least 30 min. It does not need to be high intensity for the whole period; you should strive to push yourself but not over-exert yourself.
I personally love exercise DVD's. Seriously, I couldn't live without them. I hate machines because I find them boring and often they aren't challenging enough, so I don't have gym membership and I workout from home. If the weather is horrible, I'll go running as well. I'd much rather run outside then on a treadmill.
I don't know if you have OnDemand, but if you do they have loads of free fitness videos ranging from a few minutes to an hour. Most of the workouts are beginning to intermediate levels. Netflix also has free videos.
Also, Amazon.com has tons of DVD's to choose from and they really aren't that expensive. I'd recommend trying a Jillian Micheals workout.
If you're looking to spend a little money and you are in well enough shape, then I highly recommend you try Insanity, Turbofire, Chalean Extreme, or P90x. All of those are Beachbody workouts and although they are pricey, they are absolutely amazing programs that you can do in your living room.
I just bought her 30 Day Shred and I love it!!!!
You'll need some hand weights, but that's it.
How old are you children and do you have stairs?
Lots of exercises that you can do involving your children.
1 Get on your back and put them on your bent knees. Do leg ups with them.
2 Sit with them facing you, your feet apart and their feet against yours. Hold hands and take turns pulling each other.
3 Get a ball that is their size and practice short throws.
4 Same ball but sit on the floor and roll it out as far as you can with your hands until you can go no further and let it roll to your child. They return it.
5 Play clean up. Drop stuff on the floor and both of you do squats to pick up the items.
There are more, but it gives you mommy time with your children. My grandkids love to play these games when they are over. And I had started this when my girls were young and they will still get on the floor and do them with us. I will go over to my youngest and we will throw the big yoga ball around. Might not seem like exercise but after 30 minutes your arms and legs sure feel it.
If you have stairs, start with one, go up and down. Then two stairs up and down and then three and four until you have done all the stairs. Want to make it harder go down the stairs backwards.