Regard (v.) is to consider or to have an opinion about something:
Regards (n.) is a greeting:
The phrases regarding and in/with regard to mean “related to / concerning the following topic”:
These phrases are more formal, and in everyday spoken English we’d say “about” instead: “Have you read the report about the economic crisis?”
Regardless (adv.) means one thing is not affected by something else:
As you can see from the examples, the action (keep going / approving the law / making a decision) is not affected by the other situation (the difficulties / the opposition / not having all the information).
Don’t use the word “irregardless” – it doesn’t exist; it is a common error in English.