In several missions, someone from the Fellowship of the Ring or somebody from the movies will join your party to help you defeat a certain boss or difficult enemy. These are the people that will occasionally become a fourth member of your party:

Eastern Moria - Gandalf, when fighting the Balrog of Morgoth.

East Emnet Gullies - Aranel, when fighting large groups of Uruk-hai.

Helm's Deep - Legolas, when fighting the first battle on the wall, and Gimli, when defending the wall after it has been breached.

Osgiliath - Faramir, when fighting the boss Gothmog.

Minas Tirith - Gandalf, when fighting the Witch-King of Angmar.

Pelennor Fields - Eowyn, when fighting the Witch-King of Angmar again, and Aragorn, when fighting three huge trolls and some the eight remaining Ringwraiths.