Here's a breakdown of the classes.
Sentinels- Biotics and Tech. They can tear enemy defenses apart with Overload for shields and Warp fro armor and biotic shields. Their Tech Shield also makes them the toughest class since it offers increased protection and once it's destroyed it knocks nearby enemies down giving them time to hide. The downside is their lack of weapons training limiting them to SMGs (submachine gun), pistols and heavy weapons.
Infiltrator- Combat and Tech. This class is the ideal sniper. They have Overload to deal with shields and Incinerate to deal with armor. They work best when they can snipe enemies from afar. Their Tactical Cloak renders them invisible for a time allowing them to flank enemies or better position themselves. Since they have AI Hack they can also pit mechs against enemies. However this class is obviously at a disadvantage at close range
Vanguard- Combat and Biotics. They are all about close combat. Their Charge ability lets you biotically charge enemies from a across a field and restores their shields. Combined with their access to a shotgun they are a force to be reckoned with at close range. Shockwave is good for breaking up groups and enemies and flushing them out of cover. The downside is that they have no abilities to bypass protection apart from Incendiary Ammo and are poor at long range
Adept- Biotic Specialist. This class is all about crowd control. Their Singularity ability creates a mini vortex that tosses enemies about like rage dolls and Push leaves them floating helpless. They also benefit from Warp to take care of biotic shields and armor. However this class does not do so good on the higher difficulties because their crowd control biotics don't work against protected enemies and lack long rang capabilities
Soldier- Combat Specialist. They are all about guns and ammo powers. This is the only class that can use every weapon type and the only with access to the Assault Rifle. They can dish out large amounts of damage especially when combined with their Adrenaline ability that slows down time and increases weapon damage while active. However because they are a purely combat class they don't have any special abilities
Engineer- Tech Specialist. Like Sentinels they are very good at wiping out enemy defenses with Overload and Incinerate. They are very good at keeping enemies occupied with AI Hack and Combat Drone. Their fighting style is about dishing out damage with abilities. However they lack weapon training and they can't take too much heat.
From there you should be able to deduce what squad mates work best with what class.
