I live in an apartment and I want to have some indoor trees in my apartment.
We have a large window that faces to the north-east and has curtains.
What kind of trees can be kept in an apartment?
On the assumption you mean treelike plants as opposed to smaller houseplants, have a look at these palms: Chamaerops humilis (if you have room, they spread out a bit), Rhapis excelsa (little lady palm), Phoenix canariensis and Howea forsteriana - none of these likes very hot heated rooms, they prefer things a little cooler at around 16-22 deg C.
Otherwise, provided the plant can be put somewhere out of draughts, and where its environment doesn't really change much in terms of hot/cold, Ficus benjamina or one of its variegated relatives, F. benjamina variegata/F. benjamina Starlight. The classic Rubber plant, Ficus elastica decora or F. robusta or F. Black Prince (almost black leaves), are worth considering too - there are variegated versions available (F. tricolor, F. schrijvereana). These get pretty tall but are not quite so fussy in terms of environment.
Bear in mind that all of these should only be watered when the surface of the potting medium seems dryish to the touch, that they should not be left standing in water in any outer tray or pot, should not be stood near a heat source such as a radiator, and most do not like direct sun in summer, though as your room is facing north east, morning sun will likely be all they get. Ficus benjamina varieties will tolerate a little morning sun, but prefer a bright spot without direct sunlight.