How does a vasectomy or a tubal ligation work to sterilize an individual?

Fertilization: Tubal ligation blocks the fallopian tubes preventing sperm from reaching the ovulated egg. Vasectomy blocks the vas deferens preventing transfer of sperm to the female partner during intercourse. In both cases the egg does not fertilize and an embryo does not form.

Blocked tubes: Both methods block sperm/ova from reaching their destination. In men, sperm are blocked from releasing with the seminal fluid; with women, sperm can't inseminate because there is no longer a path. Neither affect normal hormone production & should be considered permanent.

Blocked tube pathway: Sterilization by surgery involves blocking the tube between the gonad (organ with the sperms or eggs) and the outside. The surgery is very effective but failure rates are about 1-2%. About 2% of people live to regret the decision and ask for reversal. There are successes with reversal but it is different for men and women. I hope this helps clarify.