Pretty sure I've mostly seen idea as アイデア
Does anyone know of other words used for 'idea' and if possible any example sentences and contexts where different words are used?
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Use アイディア (or アイデア) when it's replaceable with resolution, suggestion, etc. If you really want to avoid loanwords, you can use "(良い)考え".
What shall we do? Do you have any idea?
An idea is something that solves multiple problems at once. -- Shigeru Miyamoto
アイデアというのは複数の問題を一気に解決するものである -- 宮本茂
Use 概念【がいねん】 (or 考え方) when it's replaceable with concept, way of understanding it, etc. コンセプト may also be a good translation, I think.
The basic idea of Darwin's theory of evolution is that all species of organisms arise and develop through the natural selection ...
Use イデア when it refers to the Plato's theory.
According to the theory of Ideas, non-material abstract forms (or ideas), and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality.
'Idea' in a sentence like this is very difficult to translate using a noun:
Do you have any idea what I am talking about?
私が何を言っているのか分かるか? (or 理解しているか?)