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Electric field lines never intersect. We know that. So closely spaced field lines mean a stronger field and field lines farther apart mean a weaker field.

How are we really sure that they never intersect? Can it be proven?

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    $\begingroup$ This is related to the uniqueness of the flow of a vector field. If two solutions of a well-behaved ODE intersect, this would lead to the violation of uniqueness. $\endgroup$ – Phoenix87 Dec 27 '14 at 22:21
  • $\begingroup$ I'm not sure this is right @Phoenix87. Electric field lines don't intersect simply because a vector field is single-valued, just like any other well-defined function. This has nothing to do with ODE uniqueness, which explains why trajectories in phase space don't intersect, for instance. $\endgroup$ – gj255 Dec 27 '14 at 22:58
  • $\begingroup$ that is the point: it is well-defined, so when you integrate it it behaves nicely. What I was proposing is a mathematical translation of this physical fact $\endgroup$ – Phoenix87 Dec 27 '14 at 23:11
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An electric field indicates the direction of the electric force that acts on a charge at that point. If the field lines ever cross, you would have two force directions, which does not make sense without combining the two directions into one direction (force is a vector), which only ends up replacing the crossed field lines with a non-crossing line anyways!

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