Use a rectangle to surround each player and enemy, the height and width of the rectangles should correspond to the object you're surrounding, imagine it being in a box only big enough to fit it.
Now, you move these rectangles the same as you do the objects, so they have a 'bounding box'
I'm not sure if Java has this, but it might have a method on the rectangle object called .intersects() so you'd do if(rectangle1.intersectS(rectangle2) to check to see if an object has collided with another.
Otherwise you can get the x and y co-ordinates of the boxes and using the height/width of them detect whether they've intersected yourself.
Anyway, you can use that to either do an event on intersection (make one explode, or whatever) or prevent the movement from being drawn. (revert to previous co-ordinates)
edit: here we go
boolean
intersects(Rectangle r)
Determines whether or not this Rectangle and the specified
Rectangle intersect.
So I would do (and don't paste this code, it most likely won't work, not done java for a long time and I didn't do graphics when I did use it.)
Rectangle rect1 = new Rectangle(player.x, player.y, player.width, player.height);
Rectangle rect1 = new Rectangle(enemy.x, enemy.y, enemy.width, enemy.height);
if(rect1.intersects(rect2))
{
System.out.println("game over, g");
}
obviously you'd need to fit that in somewhere.