A squad is composed of soldiers that can have different specialties or a single one. However, each soldier plays a specific and different role inside the squad, regardless of his specialty. One basic rule is: role and specialty can be different. For instance, a Rifleman can be employed on Assault/Support/Security roles and a Grenadier on Assault/Security roles. Specialty regards what the unit can do individually. Role defines what the soldier will do as a member of a squad or of an element of a fireteam.
Understanding this difference is an essential to successfully coordinate your squad to act as a single entity, instead of moving them individually as if you were going against the original game's AI. Whatever happens on one side of the map to the soldiers you placed there WILL affect your squad as a whole. If your right flank is killed then the rest of the squad will have to deal with the threat of enemy units moving towards them from that area. The aliens' AI might not take advantage of it since it is not programmed to recognize such advantages but an experienced player will.
This is where fireteams (also called elements) step in. On real life they help a Squad Leader to divide and to organize efficiently his squad into fighting groups which each playing a part on the overall strategy. Instead of having 9-15 soldiers acting as a mob whose sole purpose is to engage the enemy wherever it is, they make it easier to know which units should move/stay, so that the squad doesn't become unbalanced on his deployment across the map, and which should fire to concentrate the firepower in order to get kills.
Subdivisions inside a squad
The basic subdivision inside a squad is a pair of soldiers. In real life soldiers are grouped together for morale (easier to endure a bombardment if someone is close to you) and tactical reasons (one soldier fires to cover the other while it moves). On UFO2000 soldiers should be paired because a single unit can be easily killed without taking any losses (if there's 2 and the second survives....revenge! :).
Another possible division is to create a specialized weapons/support team, which is responsible to provide support to a maneuver/assault team which will be responsible to attack/defend. Another more modern deployment is for each team to be equal in order to be equally capable to support or maneuver. Or another possible one is of mass attacks where it doesn't really matter how many teams to have. In any of those cases, and with each soldier itself the question is more: what role does he play on the field.