The displayed form may substantially differ from the internal form

As soon as you discover replacement rules, you are bound to find that they mysteriously fail to replace subexpressions, or replace subexpressions you didn't expect to be replaced.

For example, consider the definition

foo = (a+b)(c+d)(e+f)/Sqrt[2]

which will cause Mathematica output an expression which looks very much like what you entered; approximately: $$\frac{(a+b)(c+d)(e-f)}{\sqrt{2}}$$ Also the InputForm seems to confirm that no transformation has been done to that expression:

((a + b)*(c + d)*(e + f))/Sqrt[2]

Now try to apply some rules on this (from now on I'll give the output in InputForm):

foo /. {x_ + y_ -> x^2 + y^2, x_ - y_ -> x^2 - y^2, Sqrt[2] -> Sqrt[8]}
(*
==> ((a^2 + b^2)*(c^2 + d^2)*(e^2 + f^2))/Sqrt[2]
*)

What is that? We explicitly requested the difference to be replaced with a difference of squares, not a sum! And why wasn't Sqrt[2] replaced at all?

Well, the reason is that Mathematica expressions are not what they look like. To see the real structure of a Mathematica expression, you can use FullForm:

foo // FullForm
(*
==> Times[Power[2, Rational[-1, 2]], Plus[a, b], Plus[c, d], 
     Plus[e, Times[-1, f]]]
*)

Now, we see why the replacement rules didn't work as expected: e-f is actually e + (-1)*f and thus matched perfectly the first rule (sum of two expressions) which transformed that into e^2 + ((-1)*f)^2 which of course evaluates to e^2+f^2. At the time the second rule is applied, the difference doesn't exist any more. Also, the Sqrt[2] in the denominator is actually a factor of 2^(-1/2). It is also easy to check that Sqrt[2] has Power[2, Rational[1, 2]] (that is, 2^(1/2)) as FullForm. That one is nowhere found in the FullForm of the expression foo evaluates to.

With that knowledge we can correct our replacement rules to work as expected:

foo /. {x_Symbol + y_Symbol -> x^2 + y^2,
        x_Symbol - y_Symbol -> x^2 - y^2,
        1/Sqrt[2] -> 1/Sqrt[8]}
(*
==> ((a^2 + b^2)*(c^2 + d^2)*(e^2 - f^2))/(2*Sqrt[2])
*)

First, we restricted our + rule to only accept symbols as expressions, so that it doesn't match e-f. For consistency, the same is true for the second rule. Finally, we replaced 1/Sqrt[2] instead of Sqrt[2] (Mathematica correctly evaluated 1/Sqrt[8] to 1/(2 Sqrt[2])).

Note that instead of FullForm you can also use TreeForm, which gives you a nice graphical representation of the internal expression).