The Feywild and the Shadowfell are terms that are tossed around quite a bit in the D&D world, but what are they exactly? I found the following description on page 8 of The Manual of the Planes:
Close by the mortal world lie its echoes, the parallel planes: the Feywild and the Shadowfell. Parallel copies are strange copies of the material world. Where seas and mountains lie in the world, similar seas and mountains exist in these parallel planes. Yet, they are not perfect copies. A thriving human city in the mortal world might be a forested vale in the Feywild and a haunted ruin in the Shadowfell. It seems that the Feywild's version of a locale in the natural world is a pristine memory of what the place was like in the youth of the world, whereas the Shadowfell's version shows a vision of how that same spot will look when its people pass away and it falls into ruin.
Although some natural features vary between these planes, the biggest difference between them are the structures of the sentient creatures. The kingdoms and castles of the Eladrin are not replicated in the mortal world or the Shadowfell. Similarly, the city of Gloomwrought exists only in the Shadowfell, and no city stands in the corresponding spot in either the mortal world or the Feywild.
The other main difference is that the senses are heightened in the Feywild. The scent of a flower - or a foul smelling bog - is stronger, colors are brighter, the sunshine is more radiant, and taste is enhanced as well. As you might expect the mood of an adventurer is heightened considerably visiting the Feywild.
The Shadowfell has the opposite effect. The mood is dark and gloomy, with a sense of pervasive melancholy. It is not a place of darkness, but rather one of shadows. Colors are grayer, food is blander, and the sun sits in a perpetual sunset or sunrise. Whereas ones mood is heightened while in the Feywild, the oppressive sense of sadness that eminates from the Shadowfell wears on even the heartiest of adventurer.
The book provides an extended introduction to both locations, going into much more detail about the environs and inhabitants of these two.