particle
Adds the meaning of 'even' to a noun or noun phrase, often implying an extreme or unexpected case. Can be used with も for emphasis (さえも).
彼は自分の名前さえ書けなかった。
He couldn't even write his own name.
そんなことは子供でさえ知っている。
Even a child knows that.
particle
Used in conditional clauses (～ば/～たら + さえ) to indicate the minimum condition needed for a result. Often translates to 'if only' or 'as long as'.
お金さえあれば、何でもできる。
If only I had money, I could do anything.
天気さえよければ、ピクニックに行く。
As long as the weather is good, we'll go on a picnic.
particle
besides; on top of that; what's more
Adds an additional, often negative, element to a situation. Commonly used with に (さえに) or in the pattern ～に加えて～さえ. Implies 'not only ... but also ...'.
雨が降っているのに、風さえ強くなってきた。
It's raining, and on top of that, the wind is getting stronger.
彼は遅刻した上に、宿題さえ忘れた。
Not only was he late, but he even forgot his homework.