Cangjie:

The most popular shape-based system by far. Cangjie is mainly used in Hong Kong, as people there often don't know Pinyin, or don't want to use Mandarin pronunciations to type. It's based on 24 different components, assigned to all of the letters of the alphabet, except for Z and X, which are used for a different purpose. In addition to its primary meaning, each component can also represent many different sequences of strokes. For example, 田 can represent itself, or it can represent an enclosing box, as in 國/国. Since there are only 24 base components, there is a lot of memorisation required to determine how to represent a character or radical. Many radicals are represented with a somewhat arbitrary code to make typing them easier, at the expense of consistency. If you decide to learn Cangjie, be prepared to look at characters very differently. For example, in Cangjie, 王 is considered to be 一 + 土.  For example, Writing the character 国, you have the code 田一土戈 (WMGI). This represents, in order, the outer box, the first stroke of 王, the next three strokes of 王, and the little drop. In a very complicated character in which the last element is 王, it might be represented as 土, because 王 is not considered to be an indivisible element, and the middle part of a long code is dropped. As a final note, Cangjie does not distinguish between certain variants of some characters. For example, 黑 and 黒 do not have separate codes in Cangjie. If you want to type in any language that uses Chinese characters that isn't Chinese written using traditional characters, Cangjie is not for you. If you plan on typing simplified characters, you need to enter the code for the traditional character anyway, and then choose the simplified one (this is at least the way thing are on the Microsoft IME - there might be a better system that can handle simplified characters better). Overall, Cangjie requires you to remember a system of decomposing characters that is quite unlike decomposing a character into its radicals. Learn Cangjie if being able to type with your preferred system on someone else's computer is important to you, or if you find it really appealing for some reason (I personally don't).

Quick:

This is just Cangjie with only the first and last letters of each code. Good if you really can't remember how to write the middle part of a character, maybe.

Dayi:

Dayi is essentially the same idea as Cangjie, but there are 46 components instead of 24. This means that you need to use the number row and punctuation keys to type characters. Perhaps the additional code space allows you to be more specific than Cangjie in selecting exactly the character that you want, but the inconvenience of this method has prevented me from learning it.

Wubi:

Wubi is essentially the simplified version of Cangjie. By that I mean that it's the shape-based input method with the most users in Mainland China. Wubi, like Cangjie, breaks characters up into components, each of which can be represented with one letter. I've read that the system is much more logical and consistent than Cangjie, but I have not used it myself, since I don't often use simplified characters. The biggest downside to Wubi is that, like Cangjie, it is focused on typing one type of characters. You can type traditional characters with Wubi, but you need to input the code for the simplified one, and then choose the traditional one as a variant. If you're only planning on typing simplified characters (which you shouldn't be - vive la résistance traditionelle!), then I have much less trouble recommending Wubi to you than I would recommending Cangjie to a traditional user. Mind you, I haven't used Wubi, but it seems pretty logical from what I've seen. Apparently it has a shared history with Zhengma (which is currently my favourite method), which makes me optimistic about its quality.

Array:

Array is another system similar to Cangjie. This time, some punctuation keys are used, but not the number row, which is nice. The big selling point of Array is that the keyboard is laid out as a grid. You can determine what component each key represents by its row and column. Presumably, it's much easier than Cangjie to learn, but not very ergonomic. I haven't used it extensively myself, but it seems to be a servicable system.

Wubihua:

Not at all related to Wubi. In Wubihua, Chinese characters are divided into five different types of strokes. All you have to do is enter which type of stroke it is for each stroke in the character. For complicated characters, this can be cumbersome, but predictions are normally provided by the software. This input method is pretty much only used on phones. My preferred last-resort method when I can't figure out how to type a character any other way.

Boshiamy:

I'd like to go in depth about Boshiamy, but unfortunately my experience with it is limited, since it's a proprietary system, and it costs money. There's a free alternative version by the community that's available for Linux, I think, but I didn't get around to trying it out extensively. As far as I can tell, it assigns 300 or so radicals to one of 26 letters. In my opinion, this might be the best system following the one-letter-per-component rule.

Zhengma:

Zhengma is my personal favourite at the moment. Unlike all of the other Cangjie-like systems, Zhengma gives each component a two-letter code (except for very common ones, which are only one letter). What I like about this is that it requires you to be quite specific about what you want, which is important for remembering how to write characters by hand, in my opinion. Note that the two-letter codes are often abbreviated to one letter when typing a complicated character, but you still need to be aware of them. Each character is represented by a code which is at most four letters long. This means that Zhengma's codes are actually shorter than Cangjie's, yet it can represent more characters. 黑 and 黒 are given distinct codes, as are most variants. Traditional and simplified versions of the same character with different radicals will have a different code. If the characters are identical except for the simplification of a radical　(example: 銜 and 衔), the code will be the same, but if you get the wrong one, just add A to the code to get the other one. Which one your IME shows you first depends on your preferences. It's just as easy to type Japanese kanji, including shinjitai, as it is to type Chinese hanzi. I think Boshiamy shares this advantage, but I'm not sure. Similar codes produce similar characters, so if you're trying to figure out what the code for a character is, it's easy to tell if you're on the right track. Unfortunately, Zhengma is one of the less popular input methods. Due to a patent dispute, Microsoft dropped it from Windows 8 and later versions. The patent now seems to be expired, but I doubt that it will be added again, as there isn't much demand for it. You can use Rime along with a custom table to type Zhengma on modern versions of Windows. On Linux, you can use IBus. On Android, there is a custom table available for Baidu's input method. Overall, I have nothing but praise for Zhengma, and I plan on continuing to learn it.