if you're looking for a quick explanation, think of it this way: the engine is like the basic situation you outline for your teacher when you dont have your homework again-- theres you, the teacher, and no homework. now a game built on this engine would still have you, the teacher, and no homework, but the teacher could be an 8-armed, 3-eyed monster with fangs; and you could be a superhero with a rocket launcher hehe. the engine is what causes you to say "hey, this game FEELS a lot like quake!". it includes how the physics, graphics, and in some cases, even the menu art is handled! look at elite forces for pc, the game basically FEELS like quake 3 (since it was built on that engine hehe) and doesnt make many improvements.
alternately, you can have a company totally improve upon an engine, to give a game a totally new feel. the benefit to the company is that they don't have to start from scratch from the ground up. the best example i can come up with is half-life -- that game was built on the quake 2 engine, but totally improved... for instance i could be wrong, but i cant recall even one use of reflection mapping in quake 2, but it is used sparingly in half life. thats one example of many on how the basic engine was improved. heh hope this helps you avoid having to read a dry paper to get your answer 🙂
*EDIT* the engine also determines how character models and object data are stored, for example a more advanced engine will allow skeletal animation systems for more fluid motion of objects, versus older methods of using keyframes to store hand-produced animations for specific parts of an objects body. ex. back in the day, the animators had to hand-manipulate the polygons in a wrist to make a hand motion; nowadays they can usually drag and drop points of the hand since its all held together by a "skeleton".