Let's talk about something exotic, hehe.
I can't speak for all Slovaks, the menu can change with region, and even the same menu differs when individual family recipes come into play.
As far as what I have known as a child:
Christmas Eve.
The big one. Gifts are unpacked on Christmas Eve, and the big dinner is on the same evening. You're supposed to fast during the day; and you're not supposed to have meat (fish doesn't count, hehe) as a part of the menu. THIS, however, is highly individual, as I know loads of Slovak people from the same city I come from who look at me like "what?" when I say this to them. My Grandma was from the north of Slovakia, so maybe that's where this comes from.
The menu would be:
1. Cabbage soup. I'm sure this came around because Slovaks were traditionally rural and poor people, so most of our traditional recipes feature cabbage and potatoes. But OMG the cabbage soup is divine, and it turns better with each day (good to make LOADS of it). Recipes vary, and what my Grandma did was to to a vegetarian one for Christmas Eve and the "real one" (with sausage and stuff) for New Year's Eve.
There are regions where lentil soup is eaten, and there are regions where who-knows-what is eaten. Still I still the cabbage soup the the most representative one.
2. Fried fish and potato salad with mayonnaise. Traditionally, the fish is carp, but we never had carp because of the bones (I was small and Grandpa was old, so we were spared). The potato salad, again, differs from family to family. Ours has potatoes, peas, carrot, apples, pickled gherkings, mayonnaise, cream, lemon juice and some seasonings.
Christmas Day.
For lunch, we used to have chicken soup and chicken as the main dish, but that varied according to taste. Maybe some people eat something traditional (duck, maybe), but we weren't too traditional in this way.