What is the Java ?: operator called and what does it do?

Others have answered this to reasonable extent, but often with the name "ternary operator".

Being the pedant that I am, I'd like to make it clear that the name of the operator is the conditional operator or "conditional operator ?:". It's a ternary operator (in that it has three operands) and it happens to be the only ternary operator in Java at the moment.

However, the spec is pretty clear that its name is the conditional operator or "conditional operator ?:" to be absolutely unambiguous. I think it's clearer to call it by that name, as it indicates the behaviour of the operator to some extent (evaluating a condition) rather than just how many operands it has.

Yes, it is a shorthand form of

int count;
if (isHere)
    count = getHereCount(index);
else
    count = getAwayCount(index);

It's called the conditional operator. Many people (erroneously) call it the ternary operator, because it's the only ternary (three-argument) operator in Java, C, C++, and probably many other languages. But theoretically there could be another ternary operator, whereas there can only be one conditional operator.

The official name is given in the Java Language Specification:

§15.25 Conditional Operator ? :

The conditional operator ? : uses the boolean value of one expression to decide which of two other expressions should be evaluated.

Note that both branches must lead to methods with return values:

It is a compile-time error for either the second or the third operand expression to be an invocation of a void method.

^{In fact, by the grammar of expression statements (§14.8), it is not permitted for a conditional expression to appear in any context where an invocation of a void method could appear.}

So, if doSomething() and doSomethingElse() are void methods, you cannot compress this:

if (someBool)
    doSomething();
else
    doSomethingElse();

into this:

someBool ? doSomething() : doSomethingElse();

Simple words:

booleanCondition ? executeThisPartIfBooleanConditionIsTrue : executeThisPartIfBooleanConditionIsFalse