The adverb たとえ, derived from 仮令(たとえ) meaning 'supposing that', is almost always paired with ても (or でも). This combination is used to introduce a hypothetical scenario in order to stress that the outcome that follows will happen regardless. たとえ is generally placed at the beginning of a sentence to establish a possibility or supposition that will be contrasted with an unchanging result.

• たとえたくさん勉強(べんきょう)しても、明日(あした)のテストは難(むずか)しいと思(おも)う。
  Even if I study a lot, I think tomorrow's test is going to be difficult.
• たとえ雨(あめ)が降(ふ)っても、明日(あした)のイベントは中止(ちゅうし)になりません。
  Even if it rains, tomorrow's event will not be canceled.
• たとえ彼(かれ)が謝(あやま)っても、私(わたし)は許(ゆる)さないつもりだ。
  Even if he apologizes, I have no intention of forgiving him.
• たとえ先生(せんせい)でも、間違(まちが)えることはある。
  Even a teacher can make mistakes sometimes.

たとえ is typically used to create a 'contrastive' condition, emphasizing that 'even if (A) is true, (B) will still happen (or not happen) contrary to expectation'. On the other hand, もし is generally used for 'resultative' conditions, where situation (A) is expected to lead directly to result (B).

• たとえ試験(しけん)に合格(ごうかく)したとしても、学費(がくひ)が高(たか)すぎるからその大学(だいがく)には行(い)けないだろう。
  Even if I pass the exam, the tuition is too expensive, so I probably won't be able to go to that university.
• もし試験(しけん)に合格(ごうかく)したら、すぐに両親(りょうしん)に電話(でんわ)するつもりだ。
  If I pass the exam, I plan on calling my parents right away.