In Fallout 3, you can have Dogmeat plus one other of the available companions. So the largest number of companions you can have is two; all but one of the possible companions besides Dogmeat have karma requirements your character has to meet before you can recruit them. Charisma in FO3 does not have any effect on your companions.
In New Vegas, there are six humanoid companions, plus two non-humanoid ones. You can have two companions at a time, one each of the humanoid and non-humanoid ones. Charisma in New Vegas affects the nerve of your companions; each point of Charisma boosts companion damage and armor rating by 5%. Karma doesn't have any impact on the ability to recruit companions, but one of them will leave you for good if your karma goes too low in that companion's company. Your reputation with the various factions in the game, however, can keep you from recruiting or keeping certain companions.