The intuition behind RS flip-flop
Lets have a look at the NOR gate version of the SR Latch:



If we start with S and R inputs low and Q high, then the bottom gate has a logic high at it's input. This means it's output Qbar is low. Since the top gate has both inputs low, it's output is high and there is no conflict.  

Now if we bring R high, Q must go low. Now the bottom gate has both inputs low so it's output Qbar must go high. If we let R go low again things stay as they are because Qbar is now high and holding the top gates other input high.  

In the animation above red is 1 and black is 0.  

If you look at it long enough it will make sense.

To clarify things a little more, here is the truth table for the above latch:



Note that both inputs high is not allowed, as it violates the Q != Qbar as discussed in the other answers.

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