Axiom of completeness:
Consumer's choice is complete. Implication: Since consumer is rational, she must have a unique preference relation. That means the consumer choice is either x1Rx2 or x2Rx1. Alternatively, consumer's choice is consistent or comparable. For unique preference relation, consumer choice must be transitive, where transitivity implies that if x1Rx2 and x2Rx3 then x1Rx3, where x3 is another commodity.
Axiom of continuity: Consumer's preference relation (R) is continuous.
Axiom of non-satiation: Consumer's choice is non-satiated in all goods. Implication: Non-satiation means larger the consumption of a good leads to larger satisfaction or utility or lower the consumption lower is the satisfaction or utility. Non-satiation of all goods (which means "goods are good" or "more is better") means any commodity bundle 'A' is preferred over another commodity bundle 'B' only if bundle 'A' consists larger quantity of at least one good and no less quantity of any other goods. Notationaly if A>B, then A is preferred over B or APB where B is any other commodity bundle.
Axiom of convexity: Consumer choice is such that indifference curve is strictly convex to the origin (i.e., utility function is quassi-concave).
Axiom of selfishness: Consumer choice is selfish.
Implication: Consumer's choice is self-guided. It is not influenced by any other consumer.