What are the differences between the races in multiplayer?

Solution 1:

Aside from the obvious class limitations, where the races differentiate the most is Evasion and Mobility, Melee and Heavy Melee Abilities, Base Health/Shielding, and their Racial Passive. All Racial Passives have increased Power/Weapon Damage and Weight Capacity bonuses for various weapon types, however, the ratios of these bonuses vary per passive.

Asari

Base Health/Barrier: 500/500
Evasion and Mobility: Asari slide on biotic energy, which uses 10% of their barrier for each use.
Melee and Heavy Melee Abilities: For their Heavy Melee, they unleash a PBAoE (Point Blank Area of Effect) biotic pulse (similar to Nova, except the charge up time is longer, as is standard with Heavy Melee attacks).
Unique Passive: Asari Justicar

Drell

Base Health/Barrier: 500/250
Evasion and Mobility: Drell do flips for evasion, during which they take 50% less damage. They also have 15% faster movement speed.
Melee and Heavy Melee Abilities: They use martial arts for their regular melee attacks, but have the standard class-based Heavy Melee.
Unique Passive: Drell Assassin

Humans

Base Health/Barrier (or Shielding): 500/500
Evasion and Mobility: Humans can do a combat roll in any direction, both in and out of cover.
Melee and Heavy Melee Abilities: Standard
Unique Passive: Alliance Training. Humans are the only race that can get a weight decrease to all weapons.

Krogan

Base Health/Shielding: 750/1,000
Evasion and Mobility: Krogan do not have an evasion mechanism and also move slower. Unlike Turians, they cannot roll from cover to cover.
Melee and Heavy Melee Abilities: Their regular melee is a headbutt, and they have a unique heavy melee ability, Krogan Charge, which lets them charge and swing their weapon, causing knockback and staggering.
Unique Passives: Krogan Berserker and Rage (replaces Fitness)

Quarians

Base Health/Shielding: 500/600
Evasion and Mobility: Quarians do a jump to the side as their strafing evade rather than a combat roll.
Melee and Heavy Melee Abilities: Standard
Unique Passive: Quarian Defender

Salarians

Base Health/Shielding: 500/600
Evasion and Mobility: Salarians do a jump to the side as their strafing evade, same as Quarians.
Melee and Heavy Melee Abilities: Standard
Unique Passive: Salarian Operative

Turians

Base Health/Shielding: 500/750
Evasion and Mobility: Turians, like Krogan, do not have any way to evade or strafe and move slower. However, they can roll from cover to cover.
Melee and Heavy Melee Abilities: Standard
Unique Passive: Turian Veteran. They are the only class that can receive a bonus to Weapon Stability.

The Resurgence Pack DLC added the Geth and Batarian races. You need to download it in order for the new races to show up in the store packs for you.

Batarians

Base Health/Shielding: 750/750
Evasion and Mobility: Batarians have the same movement limitations as Krogan. They do not have a way to evade or strafe, are unable to roll from cover to cover, and have a slightly slower overall movement speed.
Melee and Heavy Melee Abilities: Standard regular melee. They have a unique heavy melee which takes a long time to charge, but it pretty much pulverizes your target, doing massive damage, causing knockback and quite a bit of blood spatter.
Unique Passive: Batarian Enforcer. Batarians are the only race that can get a Spare Ammo capacity bonus as a passive. In addition, they can get a weight decrease to both Sniper Rifles and Shotguns.

Geth

Base Health/Shielding: 250/750
Evasion and Mobility: Geth do a strafing evade outside of cover similar to the Quarians and Salarians. Like the Krogan, they cannot roll from cover to cover. They get movement speed bonuses in their passive.
Melee and Heavy Melee Abilities: Standard regular melee. The Geth Heavy Melee is a bit similar to the Asari's. It's a PBAoE pulse that uses your shields to power it (again, similar to Nova) for as long as you hold down the Heavy Melee button or until your shields run out, whichever comes first. Higher shields will let you do this longer, so in Hunter Mode, which costs you 50% of your shields, it won't last as long.
Unique Passives: Networked AI, Advanced Hardware (replaces Fitness), and Hunter Mode. Geth can receive a weapon damage bonus to Geth weapons, and like the Batarians, can also receive a reduction in weapon weight for two types of weapons instead of one type, Assault and Sniper Rifles. Hunter Mode, when active, lets you see through smoke and architecture, in order to locate enemies, who show up with red glowing auras. There is a 50% penalty to your shields while this ability is active.

Solution 2:

Each race also has it's own variants of melee and heavy melee attacks.

Krogan are notable in this case, since their heavy melee attack is actually able to stagger Atlas mechs.

Solution 3:

Krogans also have vastly superior health and shields, though I don't remember their base stats (think its 700 health, 900 shield or something, please anyone correct me on this).