The best choice would be to use a collection, but if that is out for some reason, use arraycopy. You can use it to copy from and to the same array at a slightly different offset.
For example:
public void removeElement(Object[] arr, int removedIdx) {
System.arraycopy(arr, removedIdx + 1, arr, removedIdx, arr.length - 1 - removedIdx);
}
Edit in response to comment:
It's not another good way, it's really the only acceptable way--any tools that allow this functionality (like Java.ArrayList or the apache utils) will use this method under the covers. Also, you REALLY should be using ArrayList (or linked list if you delete from the middle a lot) so this shouldn't even be an issue unless you are doing it as homework.
To allocate a collection (creates a new array), then delete an element (which the collection will do using arraycopy) then call toArray on it (creates a SECOND new array) for every delete brings us to the point where it's not an optimizing issue, it's criminally bad programming.
Suppose you had an array taking up, say, 100mb of ram. Now you want to iterate over it and delete 20 elements.
Give it a try...
I know you ASSUME that it's not going to be that big, or that if you were deleting that many at once you'd code it differently, but I've fixed an awful lot of code where someone made assumptions like that.