What was the disagreement Socrates had with the Sophists?
Socrates, who is famous for being one of the wisest men is his day, disagreed significantly with some of the other philosophers, namely the Sophists. They believed in 'relativism', which is the idea that there is no 'absolute truth', and everyone's own idea of truth is valid.
Socrates dismissed this idea with the argument that if there was someone whose own personal truth was that relativism was not true, then who were the Sophists to tell him that relativism was true? If everyone's own view of reality is different and valid, then there could be someone who holds that not everyone's view on reality is valid, and the Sophists would not be able to argue, simply based on their own argument.
What was Plato's point in his allegory of the cave?
Plato postulated that everything here on earth, in the physical world, is a reflection of eternal forms that are outside of the physical and the mental. He sought to illustrate this with his allegory of the cave, which has a group of people living in a cave that has very little light shining into it. Outside the cave, there are objects and a great source of light that is shining upon the objects, casting their shadows into the cave. The people in the cave see these shadows and take them for reality, and that is their entire world.
The philosopher, then, is the one who leaves the cave and sees that there is more to life than the pale shadows cast into the cave, and tries to convince the people of the cave of this.
Plato's point is that what we have here on earth is all just shadows, pointing to real, eternal forms and that the true purpose of mankind is to find the forms.