1) Regardless or Irregardless
Irregardless is not really a word. It became acceptable in the American English language.
2) May I or Can
When asking someone's permission or preference, use "may I". When asking if you have the ability to do something, you use "can" in a question.
3) They're or Their
"They're" is a contraction for "they are". The word "their" shows possession.
4) Farther or Further
"Farther" refers to a measurable distance and "further" is an immeasurable distance.
5) Anxious or Eager/Excited
"Anxious" describes someone who has anxiety. "Eager/Excited" implies someone who is anticipating an event.
6) Then or Than
"Than" is used to compare two different things. "Then" is used in other instances.
7) Loose or Lose
If your pants are loose, you might lose your pants.
8) Principal or Principle
"Principal" is defined as highest in rank. "Principle" is a fundamental truth, law or standard.