The main point is that one is not allowed to cook for a gentile on Yom Tov. As a result, one should not invite a guest. A maid or waiter can be served because they will not expect extra food to be cooked for them.
May one invite a gentile on Yom Tov to eat at the Yom Tov table?
There definitely is a serious problem in inviting a gentile, on Yom
Tov, to partake of your Yom Tov meal, for the following reasons:
The possuk in (Sh’mos 12:17) states 'ach asher yochal l'chol nefesh hu
levado ya'aseh lachem', which teaches us that one may cook on Yom Tov
for the sake of Yom Tov. However, at the end of the possuk the word
לכם – for you, the gemora explains, means that one may not cook or do
any melacha for a gentile. This means that one may cook and roast
chicken for a Jew on Yom Tov, but not for a gentile.
What could be wrong, if for example the food is prepared in a single
Chazal prohibited preparing food for a gentile on Yom Tov even though
one is cooking it in the same pot without any extra effort. The
question becomes more difficult when the Shulchan Aruch HaRav 
points out that the halacha states that one may fill a pot of water
and boil it even though one only needs the amount of one cup and the
remainder will be used after Yom Tov.  (One may not say that the
excess is for after Yom Tov and it may only be filled in one
occurrence). If so, why is it ossur to cook extra for a gentile even
though there is no extra effort invested?
The answer is that when one invites a guest over to eat, one makes
sure that there is ample food for one’s household and for the newly
arrived guest. Chazal were afraid that one would cook extra food for
the guest in a separate cooking pot and thus violate the issur of
cooking for a gentile. Therefore, even though one does not intend to
cook in a separate pot for the gentile, there is reason for concern
that it might lead thereto. 
Does that mean that I may not give my Yom Tov food to my gentile maid?
No, it is not the same. The reason for the distinction is because
one’s maid is not a “guest” and is not treated with the same honor.
One would not cook extra food in a separate pot for the maid and
therefore there is no room for concern. However, one may only add
extra to the pot in the first place before the initial cooking but not
add to the pot after it is already on the stove even for one’s maid.
There are exceptions to the rule, for example, certain dishes have a
better taste when there is more food in the pot, which would permit
adding food to the pot even after it is on the stove, but a rav would
need to be consulted because this does not always apply. As mentioned,
the optimal is to add as much as necessary to the pot before cooking.
May I invite the guest on Yom Tov after the food is prepared? It seems
that then there is no room to be concerned that I will cook more food.
Even in this case it is forbidden to invite a gentile. 
What if the gentile arrives uninvited?
If the guest arrives after your meal is already prepared the person
may partake of the meal. If however the person is a VIP, there are
those who prohibit this unless you explicitly say to the guest that he
may share whatever you have prepared. Even if the person arrives of
his own accord, you may not coax and invite the guest, because that is
similar to an invitation. 
 Simon 512:3.
 Shulchan Aruch HaRav simon 503:4, Mechaber 503:2 and M”B 14-15.
 Shulchan Aruch HaRav simon 512:3.
 M”B simon 512:3. See the Sha’ar Hatsiun 3 who argues with the M”A
who permits this, because from the Rambam one sees that even this is
 M”B simon 512:10.