Operators in Java include unary, binary, and ternary operators that perform operations on operands. Binary operators like arithmetic, relational, and logical operators follow an order of precedence unless overridden by parentheses. Statements in Java include if-then, if-then-else, switch, while, do-while, for, and more to control program flow. The if-then statement and if-then-else statement evaluate conditions and execute code blocks conditionally. The switch statement compares a value to multiple case labels and executes the corresponding code block.