もと

“Origin”

もと is a simple, yet really broad word. It means “Origin” but in practice it can mean:

• Base (Physical)
• Start (Time, or of an object)
• Under (Physically or influence)
• “Around” (From physical base or under meaning).
• Original Amount
• Original Thing or material
• Original Cause
• Origin of X 
• Originally (Former) Prefixes to a noun. 

For simplicity, you can break these meanings into 2 categories:
Origin: Time, form, origin point, former, and cause; and
Base: Physical base, around the base, and under

Please read the examples to get a better idea of how broad this word is. 

See the bottom for Kanji information, but かな is ok. 

Examples

• 煙の元 – Origin of the smoke
• 木の本 – Base of a tree
• 彼の元に集める – Gather around him. 
• 親の元から去っていく – Leave their parents. (From around their parents/base of parents)
• 病気がもとで死ぬ – With sickness as the cause, died. 
• 警備員が元警官だ – The guard is a former (originally) cop. 
• 一撃のもとに倒した – Defeated with one blow (Cause). 
• 国の基を築く – Create the basis for a country. 
• 法律の下でできない – Can’t do under the law (Cause of the law). 
• 昼の太陽のもとでは働けない – Can’t work under the noon sun. 
• 本を元に戻す – Return the book to its original position.  
  
  

のもとで

“Under”
Physically/Learning/Cause/Influence

のもとで is a phrase using もと. This phrase typically means “under“, but could be any of the meanings above.
It usually refers to being under physically, under the influence, or learning under someone (Not socially).

Examples

• 先生のもとで学ぶ – Learn under the teacher.
• 月光のもとで進む – Advance under the moonlight.
• 親のもとで暮らす – Live under/with your parents. (Around/influence)
• 規則の元で罰する – Be punished by/according to/under the rules. 
• 酒の元で首になった – Got fired due to alcohol. 

As a Suffix

Near/Around

Noun + もと as a suffix tends to mean “near” or “around”. Usually written as 元

Examples

• 手元にある本。- Book that is on hand/Near my hands.
• 足元に注意 – Watch out around your feet.
  
  

As a Prefix

Former/Originally a

When もと comes before a noun like a prefix, it means “Former” or “Originally”.
Usually written as 元, may be written as 旧(Old/long ago)

Examples

• 元警官 – Former cop
• 元先生 – Former Teacher
• 元学生 – Originally a student. 

See also に基づいて

Kanji

Skippable.
There are honestly too many different Kanji that can be used for もと.
I’ll try to summarize them here.

Common

• 元 – Source. Former. Near. Start. (Use this one generally). 
  • 光の元を消す – Put out the source of the light. 
• 下 – “Under” “Bottom” “Influence”
  • 木の下 – Under the tree.
• 本 – Past origin.
  • 問題の本 – Origin of the problem
• 基 – Physical base or basis
  • 彼の意見を基に進む – Use his opinion as the basis to go forward.

Situation Specific

Don’t go out of your way to use these. 

• 旧 – Sometimes used in place of 元 for “Former”. More common with place names. Also pronounced きゅう
  • 旧玉木亭 – Formerly known as Tamakitei. 
• 許 – “Permission” or “Approval”. 元 is acceptable
  • 先生のもとで勉強する – Study under the teacher.
• 素 – Ingredient or material that something was made from.
  • 小麦粉と砂糖を素で作った – Used flour and sugar to make it. 

Source