Learning. Like in most countries, children in Japan, China, Taiwan and Korea start learning how to write their language at the age of 6, reaching the mature level (i.e. the necessary level to read the newspaper, which is 2,000 kanji in Japan, 6,000 to 8,000 in China) around the age of 16. In these countries, the learning process mostly consists of brutal memorization and repetitive exercises. Children write hundreds of lines of the same character, for their hand to memorize it.
The adult foreigner, on the other hand, does not have 10 years to learned those characters, does not have the memorization abilities of a 6 years old and does not necessarily have the time to spend hours writing the same character. The learning approach needs to be more rational with some way to logically link the character, its pronunciation and its meaning.

Processing. Computers and other processing machines that are now part of our everyday life, need to deal with a set of a virtually unlimited number of characters. This includes printing, keyboard input, automatic hand-written recognition, OCR (Optional Character Recognition)... Many of the western approaches for addressing these issues and provide an efficient human interface to these processing machines do not apply to almost unlimited character sets (a typical example is the keyboard input, where it would not be conceivable to have a 10000-keys keyboard). Just as a little anecdote, automatic hand-written recognition is easier to develop with kanji than Latin characters, only because kanji characters require a consistent writing style (and the machine can compute a character using stroke order, stroke count, and shape of the hand written drawing of the character), where Latin characters do not (and the machine is left with the shape of the hand written drawing of the character only to compute the correct character).

By Radical.

By Frequency of Use. Some kanji are used very often, some less often and some almost never. For the student, it is better to start with the popular kanji, and get into the less used ones later on. Based on statistics made recently, each kanji in use has been attributed a frequency of use (which is of course different in China, Japan, Taiwan and Korea). From these statistics derive the order in which kanji are traditionally learned in Japanese and Chinese schools. For more details on kanji frequency, I have a complete page (mainly Japanese kanji) about it here.

By Stroke Count.

By Stroke Order

By Shape. 4 corner code...