'a' is of type char which - coming from a C tradition - can be interpreted as a number type. That is, you can do arithmetic with it just like with int or long.

The value of a char is traditionally given by the ASCII table where 'a' happens to be the same as 97.

In your code, c is also a char, and if you assume s1 and s2 to only contain lower-case letters, you know that c is somewhere in the range [97,122]. Subtracting 'a' (i.e., 97) from that then gives you the more convenient range [0,25] to work with.

I say "more convenient" because here, c - 'a' is used to index the array a, and as you know, array indices start at 0.

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 

Postfix Operation:

a++ or a-- is postfix operation, meaning that the value of a will get changed after the evaluation of expression.

x = a++;
// This can be rewritten as
x = a;
a = a+1;

// Similarly
y = a--;
// is equivalent to
y = a;
a = a-1;

Prefix Operation:

++a or --a is prefix operation, meaning that the value of a will get changed before the evaluation of expression.

x = ++a;
// This can be rewritten as
a = a+1;
x = a;

// Similarly
y = --a;
// is equivalent to
a = a-1;
y = a;

Example:

lets assume this;

a = 4;

b = a++; // first b will be 4, and after this a will be 5

// now a value is 5
c = ++a; // first a will be 6, then 6 will be assigned to c

Refer this answer also.