If two people are eating a meat meal with bread and a third is also eating a meal with bread, but with dairy or soft cheese, they can join together to make Zimmun (the invitation to recite grace after the meal). The reason is because the one eating the soft cheese could eat from the same loaf of bread as the others, as long as he cleans his teeth properly beforehand.
Once he does this he can eat from their loaf, even if it has meaty grease on it, since he would be permitted to eat actual meat. The Mishnah Berurah writes that the custom is that the one eating the cheese is the one who should lead the Birkath Hammazon.
This applies to regular dairy or soft cheese. Hard cheese is different. Since the custom is not to eat meat right away after eating hard cheese, then even if one cleans one's mouth and teeth thoroughly, the one eating the hard cheese cannot eat from the same loaf as those eating meat. As such, he may not join together with them to recite the Zimmun.
(See Shulhan 'Arukh with Rama, 196:3. Kaf Hahayyim, ibid., Oth 11. Mishnah Berurah, ibid., 9)
Zimmun if some ate meat and some ate dairy, Birkat Hamazon,