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The dots are raindrops. The 巾 looking thing resembles the act of falling from the clouds, and the top line symbolizes the heavens. This is simple enough. Most pictographs are thankfully still somewhat straightforward. 

オンよみ： ウ
くんよみ： あめ、あま-、-さめ

The オンよみ is used in compounds excluding native compound words. Usually you can tell the difference because the former are usually either more technical or have an easier/native way of saying it whereas the native words are not so technical. あめ means rain and is homophonous with candy. In standard accent the pitch is on the first mora for rain. あま- is a sound change in the majority of native words that start with it. -さめ is likewise as a suffix.

Example Words

工 means work. The rest is the left hand. This character means "left". 

オンよみ: サ
くんよみ: ひだり

Example Words 

Be careful. The initial stroke is the vertical one. This change distinguishes it as the right hand from the left hand seen previously. 口 is a mouth. It used to mean verbal support, but this has been given to another character based off of it, 佑. So, it's an example of a rebus character. Since most people put things in their mouth with their right hand, it eventually became used to mean right hand and then simply to just right.

オンよみ: ウ、ユウ
くんよみ: みぎ

The first reading is usually seen when it is the first character in a compound. Then, ユウ is typically used when its the second. However, ユウ can still be used in the front of a word to show the original meaning of assistance. Thus, you can see how the history and development of a character goes hand in hand to why certain readings are assigned. The different オンよみ, of course, come from different stages of the Chinese languages. Thus, when new characters were introduced, so were new readings and developed meanings from the mainland. 

Example Words 

The button is a mouth and a tongue with the symbol for stand, 立, on top to symbolize sound. 

オンよみ: オン、イン、ノン
くんよみ: おと、ね

Usually, オン is the most common オンよみ. ノン　is a sound change seldom used when the previous character starts with a ん. イン, on the other hand, is often seen when the second part of a compound. However, there are many words made by using the first and foremost reading as a suffix for sound. The difference between おと and ね is that the first means "sound/noise" and the latter is more specific to "tone". 

 Example Words

　　　　5 used to just be five lines. Alas, a thread-like pictograph has always been a substituted and eventually reached its current form. 

オンよみ: ゴ
くんよみ: いつ-

 Example Words

This came from a clenched fist, which was an apparent means of expressing six in ancient China. A much more helpful interpretation is looking at the button, which means eight (八), and the top and doing 8 minus 2.

オンよみ: ロク
くんよみ: む（つ）

Example Words

This character was favored long ago for seven for phonetic reasons due to its simplicity. 

オンよみ: シチ
くんよみ: なな、なな（つ）、なの-

なの-, rarely seen as なぬ-, is used in the word for the 7th day of the month, 七日.

  Example Words

This was used to represent eight because it is easily divisible, which makes sense given the shape of the character. 

オンよみ: ハチ
くんよみ: や、や（つ） 

 Example Words

This is a pictograph of a bent elbow and was a means of expressing 9. Thus, it means 9. 

オンよみ: キュウ、ク
くんよみ: ここの（つ）

キュウ is generally preferred because ク is homophonous to 苦 (suffering). However, there are instances where only one of the readings is acceptable.  

 Example Words

The symbol for ten, 十, came from a pictograph of a sewing needle and was used as a substitute for the more complex character 拾. The original character for "ten" is still used on currency. 

オンよみ: ジュウ
くんよみ: とお

Example Words 

The character for woman comes from a pictograph of a kneeling woman with arms stretched forth. Obviously we can see what men have thought of women for centuries...

オンよみ: ジョ、ニョ、ニョウ
くんよみ: おんな、め

ジョ is the most prevalent in compounds. The others are now seen only in words that are no longer political correct due to demeaning nuances towards woman. As for native words, め is in a lot of compounds. おんな is the word for woman. 

 Example Words

Notes:

1. 女人 is basically no longer used. 女性 is the most politically correct word for woman. 女房 is not the most preferred word for wife. 妻（つま） is.

2. あなた is almost never written in 漢字, but when it is, you have several options.  

　　　This character is composed of the characters for strength 力 and field 田. It is used to mean "man". It sounds logical enough.

オンよみ: ダン、ナン
くんよみ: おとこ、お

ダン is generally used in compounds, and ナン is generally used to indicate the order of birth of sons. おとこ means "man". お is a rather elegant reading seen in some words and names.

 Example Words 

　　　　The original form of this character was 國. 或 shows an area and the surrounding radical is an enclosure. So, this gives the meaning of a country/region. It used to refer to individual domains in feudal Japan. Now it is used in the full sense of country or region in some words. The complicated part got replaced with jewel 玉. Many people wonder why during simplification king 王 wasn't used instead.

オンよみ: コク
くんよみ: くに

Example Words

This is a pictograph of someone trying to make themselves look as big as possible. 

オンよみ: タイ、ダイ
くんよみ: おお（きい・きな）

ダイ is the most common, but both ダイ and タイ are common in compounds. ダイ can be used as an independent word to mean large (size). It also doesn't help that both ダイ and おお can be used as prefixes. It's really best to learn on an individual basis for this character.

 Example Words

This is from a pictograph of three small dots. These small dots symbolize being small. 

オンよみ: ショウ
くんよみ: ちい（さい・さな）、こ-、お-、さ-

The difference between こ- and お- is that the first is used to show something is less in size or quality while the latter shows that something is nice/little. さ- is seen in some words to show elegance. 

 Example Words