Java 8+

Use String.join():

String str = String.join(",", arr);

Note that arr can also be any Iterable (such as a list), not just an array.

If you have a Stream, you can use the joining collector:

Stream.of("a", "b", "c")
      .collect(Collectors.joining(","))

Legacy (Java 7 and earlier)

StringBuilder builder = new StringBuilder();
for(String s : arr) {
    builder.append(s);
}
String str = builder.toString();

Alternatively, if you just want a "debug-style" dump of an array:

String str = Arrays.toString(arr);

Note that if you're really legacy (Java 1.4 and earlier) you'll need to replace StringBuilder there with StringBuffer.

Android

Use TextUtils.join():

String str = TextUtils.join(",", arr);

General notes

You can modify all the above examples depending on what characters, if any, you want in between strings.

DON'T use a string and just append to it with += in a loop like some of the answers show here. This sends the GC through the roof because you're creating and throwing away as many string objects as you have items in your array. For small arrays you might not really notice the difference, but for large ones it can be orders of magnitude slower.