Assuming that this is a learning exercise, I wouldn't spoil your fun by writing your code.
Here is a simple algorithm that needs only one explicit loop: go through all suffixes of the first string, and check if they represent a prefix of the second string. Start with the longest suffix (which is the entire first wors) and continu with the shorter and shorter ones. Once you located a suffix that matches the prefix of the second word, remove that suffix, and append the second string to the result.
In Java you get a suffix of a word by calling substring with one parameter. To check if a string is a prefix of another string, use startsWith method.