You ask:
If you do have [a bijection between morphisms FA → B and morphisms A → GB], then could you not also take your bijection 'in the other direction' between morphisms GA → B in C and morphisms A → FB in D [...]?
This argument doesn’t quite work, since taking the bijection in the other direction, it will go between morphisms A → GB and morphisms FA → B. The functor G still only appears on the codomains of morphisms.
To give a concrete example where no such bijection exists: let (F, U) be the “free group”/“underlying set” functors between Set and Gp.
Now, F is left adjoint to U. But they can’t be adjoint the other way round! If they were, that would give a bijection between $\mathbf{Sets}(U1,\phi)$ and $\mathbf{Gp}(1,F\phi)$ (where $1$ denotes the trivial group, and $\phi$ the empty set). But $\mathbf{Sets}(U1,\phi)$ is empty, while $\mathbf{Gp}(1,F\phi)$ has one element, since $F\phi \cong 1$.
An intuition for adjoints? There’s no easy, one-size-fits-all answer; but a good place to start is with these sorts of free/forgetful examples. Typically one can thing of a left adjoint as adding stuff, as freely as possible — perhaps new elements, perhaps some structure, perhaps imposing some equations if necessary. On the other hand, a right adjoint typically forgets things — forgets structure, sometimes perhaps throws away elements too...
The point you mention that they're a generalisation of inverses is also a good one. The Stanford Encyclopedia of Philosophy calls them “conceptual inverses”, which depending on how you feel about philosophy may be very helpful or not at all.
But really the best way to get intuition for adjoints comes from looking at as many examples as possible; not necessarily all in a hurry, but every now and then, for a while. They’re one of those concepts that doesn't usually come quickly (it didn't for me, nor for anyone I've seen learning category theory), but which — if you give it time to percolate, and occasional exercises — will sooner or later “click” and suddenly seem so natural you can't imagine not understanding it.
Incidentally, a similar question was also asked some time ago at mathoverflow. I very much like the current second answer, giving a rather different example of adjoints: viewing the posets Z and R as categories in the usual way (i.e. there's a unique map x → y whenever x ≤ y), the “ceiling” function R → Z is left adjoint to the inclusion Z → R, while dually the “floor” function is right adjoint to the inclusion. This suggests the slogan: left adjoints round up (again, adding just as much as is needed to get an integer); right adjoints round down (forgetting the non-integer part).
To add to all the answers above, there is a delightful example in the text "Mathematics for Physics" by Stone and Goldbart, (Appendix A.3), to clarify the difference between vectors and co-vectors, which I can't resist quoting here.
One way of driving home the distinction between $V$ and $V^*$ is to consider
the space $V$ of fruit orders at a grocers. Assume that the grocer stocks only
apples, oranges and pears. The elements of $V$ are then vectors such as
$x = 3 \text{ kg apples }+ 4.5 \text{ kg oranges } + 2 \text{ kg pears.} $
Take $V^*$ to be the space of possible price lists, an example element being
$f = (\$3.00/\text{kg}) \text{ apples}^* + (\$2.00/\text{kg}) \text{ oranges}^* + (\$1.50/\text{kg}) \text{ pears}^*$
The evaluation of $f$ on $x$
$f(x) = 3 \times \$3.00 + 4.5 \times \$2.00 + 2 \times \$1.50 = 21.0$
then returns the total cost of the order. You should have no difficulty in
distinguishing between a price list and box of fruit!
Best Answer
A simple arithmetic isomorphism: in many (older) video games, points are given out in multiples of $1000$ (say) to create a sense of excitement. We could scale down the points by a factor of $1000$ and preserve the essential structure of the point system: for example, if you finish the game and get the highest score, you would still get the highest score in the scaled-down version. Specifically, the scaled-down final score is the sum of the scaled-down points during the game.
In planar geometry, consider that diagrams and proofs on a piece of paper don't change when you rotate that paper through any angle in three-dimensional space or carry it around the room.
There are strategic situations that we refer to as rock/paper/scissors because they can effectively simulated with that game, the only difference being the labels of the strategies.
The Richter scale is an example of the isomorphism between the reals under addition and the positive reals under multiplication.