Here is the quick and dirty description of the greenhouse effect using a greenhouse (what else?) as an example.

A greenhouse works because of the glass panels that line the roof and walls. The glass is transparent to the visible light from the sun, so sunlight can shine in and warm things inside the greenhouse. Now a body at about 35°C emits mostly infrared radiation. (On the other hand our sun, with a surface temperature of about 5500°C, emits mostly visible light.) The glass panels are opaque to infrared light. The result is that the glass lets the energy of the sun in, but won't let it back out. This keeps the inside of a greenhouse warm.

Now to show why the greenhouse effect is a very good thing here are examples from our solar system.

Mercury

Mercury is the closest planet to the sun, so you would expect it to be hot.

It is, but only on the day side.

During a Mercutian day temperatures reach 464°C, but because there is no atmosphere to cause a greenhouse effect, temperatures drop to -183°C at night. That's cold enough to warrant an investment in triple-paned windows.

Earth

Our greenhouse effect keeps our night side almost as warm as our day side, allowing life to flourish and keeping the world's collective heating bill manageable.

But Now the Bad News

Venus

Venus is an example of having a greenhouse that is too good at trapping in the heat.

Even though Venus is twice as far away from the Sun as Mercury, Venus is the planet with the hottest surface temperature day or night. At about 484°C, lead melts on the surface at any time of the day.

The reason why this doesn't happen to Earth is due to the fact that Venus has a much thicker atmosphere with many different kinds of gases. Earth's atmosphere is full of gases that trap infrared radiation (so-called greenhouse gases), but they only trap certain wavelengths at a certain efficiency. In other words, a lot escapes. Venus's atmosphere is more diverse as well as more dense, so it traps more wavelengths of infrared radiation at a greater efficiency, making the surface hellishly hot.

That is where the greenhouse effect gets its bad reputation. We do not have a lot of carbon dioxide in our atmosphere right now, and if more is released into the atmosphere, more of the infrared radiation that it traps will be held in, making the world warmer than it is. This kind of global warming is a bad thing.