They both apply, reducing damage from attacks the other doesn't stop.
So, against an evil attack, the DR/good applies (reducing 5 points), while against a good attack, the DR/evil applies (reducing 10 points). Against non-evil, non-good attacks, you would apply the strongest DR (also reducing 10 points).
They overlap, so you apply whichever is the most effective against a particular attack.
So, an attack that counts as neither good nor evil will be reduced by 10.
An attack that is good, but not evil, would also be reduced by 10.
An attack that is evil, but not good, is only reduced by 5.
An attack that is both good and evil (from a +5 weapon, for example) would not be reduced at all.
Edit: Ninja'ed.
Yes :)
Essentially, yes. Note, however, that that is still distinctly different from "DR10/good and evil", which would reduce all attacks by 10 unless they counted as both good AND evil.
How is that "distinctly different"?
He's probably pointing it out because some people might believe "DR 10/good and evil" would be the result (according to questions stated in past threads), rather than "DR 5/good and DR 10/evil" :)
"DR10/good and evil" requires that the attack satisfy both requirements to bypass the 10 damage reduction. So, a weapon that is only good or only evil still has its damage reduced by 10.
Meanwhile, having "DR5/good" and "DR10/evil" adds a third state -- you bypass one of the types of damage reduction, but not the other. If you don't bypass either, only the largest applies, so a normal mundane weapon has its damage reduced by 10 (rather than adding the DR together to reduce it by 15). If you bypass one, but not the other, only the one that you do not bypass applies. So an evil weapon has its damage reduced by 5, since it does not bypass "DR5/good". Compare that to above, where the evil weapon would still have its damage reduced by 10, because it doesn't meet the requirements to bypass the damage reduction at all.
"DR10/good and evil" requires that the attack satisfy both requirements to bypass the 10 damage reduction. So, a weapon that is only good or only evil still has its damage reduced by 10.
Meanwhile, having "DR5/good" and "DR10/evil" adds a third state -- you bypass one of the types of damage reduction, but not the other. If you don't bypass either, only the largest applies, so a normal mundane weapon has its damage reduced by 10 (rather than adding the DR together to reduce it by 15). If you bypass one, but not the other, only the one that you do not bypass applies. So an evil weapon has its damage reduced by 5, since it does not bypass "DR5/good". Compare that to above, where the evil weapon would still have its damage reduced by 10, because it doesn't meet the requirements to bypass the damage reduction at all.
This is correct. From the SRD: "If a creature has damage reduction from more than one source, the two forms of damage reduction do not stack. Instead, the creature gets the benefit of the best damage reduction in a given situation."