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I see references to Lagrangian Submanifolds in the literature but don't know what they are. Is there a relation to Lagrangian Tori (which I also don't know what are). Could someone give a definition that includes an intuitive explanation for what their use is?

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    It's not entirely clear from your question whether you found these definitions: http://en.wikipedia.org/wiki/Symplectic_manifold#Lagrangian_and_other_submanifolds and http://planetmath.org/encyclopedia/LagrangianSubmanifold.html .2011-04-11

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It's a special kind of submanifold of a symplectic manifold; see the definition on Wikipedia. There was a question about this on Math Overflow not long ago, where you will find some good answers.

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    @joriki: Hello again! No need to apologize. Writing that explanation must have been more work for you than writing the comment in the first place. ;)2011-04-11