By Rabbi Dovid Ostroff
Three people who have dined together are obligated to bentch with zimun. This obligation is biblically sourced in the possuk âãìå ìã’ àúé åðøåîîä ùîå éçãéå, speak of Hashem‘s greatness with me, which implicates that one person is talking to at least two people, and together we will appreciate His name’s greatness.
Indeed several poskim learn that the chiyuv is biblical, while others learn that it is a rabbinical obligation, relying on a possuk for a source. Regardless, it is an important mitzvah and the Shulchan Aruch  does not mince words stating that one is chayav to do zimun.
What comprises zimun?
The essence of zimun is to say ðáøê ùàëìðå îùìå, let us bentch to the One who we ate from and the others respond áøåê ùàëìðå îùìå åáèåáå çééðå, boruch is the One who we ate from and we live because of His goodness. 
So why do we say øáåúé ðáøê?
The Magan Avraham cites the Zohar saying that one should verbally summon holy actions to embed them with kedusha.  Consequently we say, gentlemen, let us give praise. After preparing and inviting others to zimun we say ðáøê ùàëìðå, which is the actual zimun.
What are the prerequisites for zimun?
Several factors must exist to necessitate zimun, 1- eating together, 2- type of food, 3 – eating in one place, 4 – place of eating, 5 – who joins the zimun.
Three people who dined together must do zimun. Dining together is defined either by beginning to eat together or ending together. 
If two people dined together and a third joined later: if the two finished eating and the third is still busy eating, they need not wait for the third to complete, they can bentch without zimun. If however they concluded their meal and have not as yet bentched and are at a point where, if offered more food to eat they would eat it, it is not considered as if they have concluded their meal, but rather as if all concluded simultaneously and are obligated to bentch. 
Type of food
Three people who consumed bread are obligated to zimun.
What should the third person eat to complete zimun?
The Shulchan Aruch  cites various opinions. The first opinion – he should consume a k’zayis of bread.  Second opinion – a k’zayis of mezonos, something for which one recites òì äîçéä.  Third opinion – a k’zayis of any vegetable or food item. 
The Shulchan Aruch consequently rules that if the third person will not eat bread, it is preferable not to offer him a beverage or any food so as to avoid the machlokes. However, the Mishna Berura rules  that if he does not wish to wash for bread, custom is to offer a drink or any food item. 
Eating in one place
A prerequisite is that all three people dine in one place as a ÷áéòåú – a get together. The poskim define this as eating at the same table or on the same tablecloth (picnic?). The Vilna Ga’on holds that members of the same family are called a ÷áéòåú even when dining on separate tables. Milling around is not a ÷áéòåú.
People who would like to eat together, but for lack of place are eating on separate tables, are called a ÷áéòåú and may do zimun. Consequently, diners at a wedding even though they are sitting at separate tables are called a ÷áéòåú, because in essence they share the same simcha and are eating together.
A yeshiva dining hall is the same. Nobody would object to placing all the tables together and eating at one long table, because it is for sake of keeping order etc. that tables are separate. If however, each shiur wants to specifically sit separately from another shiur, they are not obligated to zimun.
On the other hand, diners in a café, restaurant or hotel who are sitting at separate tables prefer to dine alone and not sit with other people, and as such they are not obligated to zimun.  If the same waiter serves several tables, the diners at those tables are considered eating together.
Are people dining in a car or bus considered a ÷áéòåú?
The Magen Avraham has his doubts,  because halachically travelers are likened to people walking (øåëá ëîäìê ãîé) and perhaps they are not called a ÷áéòåú. When in doubt, the poskim say one may perform zimun, but in presence of ten people, since Hashem‘s name is mentioned, one may not utter Hashem‘s name.
 Siman 192:1.
 See M”B 192:2.
 Siman 193:2.
 See M”B 193:19.
 Rambam chap. 5 Hilchos B’rachos.
 Rif and Rashba.
 Tosefos, Rosh and R’ Yonah.
 M”B siman 197:22.
 As for Sephardic custom, see åæàú äáøëä page 130.
 See åæàú äáøëä page 133.
 See M”B 167:62.