Adding binary numbers

Use Integer.parseInt(String, int radix).

 public static String addBinary(){
 // The two input Strings, containing the binary representation of the two values:
    String input0 = "1010";
    String input1 = "10";

    // Use as radix 2 because it's binary    
    int number0 = Integer.parseInt(input0, 2);
    int number1 = Integer.parseInt(input1, 2);

    int sum = number0 + number1;
    return Integer.toBinaryString(sum); //returns the answer as a binary value;
}

Martijn is absolutely correct, to piggyback and complete the answer

Integer.toBinaryString(sum);

would give your output in binary as per the OP question.

To dive into fundamentals:

public static String binaryAddition(String s1, String s2) {
    if (s1 == null || s2 == null) return "";
    int first = s1.length() - 1;
    int second = s2.length() - 1;
    StringBuilder sb = new StringBuilder();
    int carry = 0;
    while (first >= 0 || second >= 0) {
        int sum = carry;
        if (first >= 0) {
            sum += s1.charAt(first) - '0';
            first--;
        }
        if (second >= 0) {
            sum += s2.charAt(second) - '0';
            second--;
        }
        carry = sum >> 1;
        sum = sum & 1;
        sb.append(sum == 0 ? '0' : '1');
    }
    if (carry > 0)
        sb.append('1');

    sb.reverse();
    return String.valueOf(sb);
}

You can just put 0b in front of the binary number to specify that it is binary.

For this example, you can simply do:

Integer.toString(0b1010 + 0b10, 2);

This will add the two in binary, and Integer.toString() with 2 as the second parameter converts it back to binary.