Wood is outdoors anyway so any real wood will be fine for a shed. The most important thing is to ensure that the wood is treated properly to protect it from rot. Man-made woods, such as MDF, are not suitable for a shed, even after treating, as sheds are constantly at the mercy of the weather. Other woods such as oak and teak are not used often, as they are more expensive.
Pine is the most common wood used for outdoor construction, as it is inexpensive, sturdy and easily treated. An added advantage is that its appearance can be altered by treating it, as pines takes well to colored paints.
This is normally only used on more-expensive sheds, but is a good all-round material. It is not usually colored, as the beauty of it is in the grain and color of the wood itself, but it will still need to be treated.
Cedar is the middle ground between pine and redwood. It is not as expensive as redwood, but is also normally treated with only oil so it maintains a wood grain finish.