さえ

Even
Noun + さえ

さえ is a particle meaning “Even X”
It emphasizes some number/noun as “surprising” in terms of the verb.
Its essentially a stronger version of も

さえ may become さえも especially with negative meaning sentences.
It just adds extra emphasis to “even”.

Examples

• 先生さえわからない
  Even the teacher doesn’t know.
  
  
• 自分の名前さえも忘れた
  Even forgot their own name.
  
  
• 気づきさえもしなかった
  Didn’t even realize 
  
  
• チャンスさえも与えなかった
  Didn’t even give a chance.
  
  
• 子供さえできることだ
  Something even a child could do. 
  
  

でさえ

(Not) Even with/by X
Method/Person+ でさえ + Verb (Usually negative)

でさえ is an expression, but its meaning isn’t really surprising:
It is just さえ coming after the で particle. Its not really different from さえ, except it is used mostly in negative sentences.
It marks a way/time/how/etc. that is surprising or extreme. Typically used for negative sentences.

Examples

• かれでさえできない
  Not even he can do it.
  
  
• 彼は夏でさえ靴下をはく (Weblio)
  He wears socks even in summer.
  
  
• 車でさえ動かせない物体
  A thing that not even a car could move. 
  
  
• 専門家でさえもわからないことだ
  Not even an expert knows. 
  
  

さえ＋ば

Minimum Condition
Noun ＋ さえ + Verb ば form
Verb ます Stem + さえすれば
Verb て Form + さえいれば

さえ is also frequently used in conjunction with the ば conditional forms of verbs.
This pattern expresses: “If only we could do X”.
It is used to express a minimum requirement or condition for the rest of the sentence. 
It has to be ば because the stress is on the action. 

Examples

• 金さえあれば何とかできる
  If only we had money, we could do something
  
  
• 力さえあれば。。
  If only we had power.
  
  
• 彼さえこれば大丈夫だ
  If only he would come then it would be ok.
  
  
• 彼女さえいれば幸せになれる
  If only I had a girlfriend I could be happy.
  
  
• この薬を飲みさえすれば、楽になる
  If you just take this medicine, you’ll feel better.
  
  
• 忘れてさえいなければ
  As long as you haven’t forgotten.
  
  
• 宿題さえすれば合格できる
  As long as you do your homework, you’ll pass.
  
  
• 謝りさえすれば許される
  You’ll be forgiven as long as you apologize.
  
  
• 人さえいなければ大丈夫
  As long as no people are there its ok. 

On top of that

I wasn’t kidding when I told you it was essentially the same as も.
Although rare, it can also be used as も (also) in some sentences.
It has more of a “on top of that” or “It even then..” meaning though.
Commonly used with だけでなく

Examples

• 風が吹き出していただけでなく、雨さえ降りだした (Weblio)
  Not only was it windy, it even started raining. 
  
  
• 車がガス欠になっただけじゃなく、タイヤさえもパンクした
  Not only did my car run out of gas, the tire also punctured.