What bonusses does a Roman Empire and a Roman Republic give

As mentioned by @5pike, the red skull symbolises attrition.

There are several reasons why an army or a fleet can suffer from attrition, here are some:

• Your army is currently laying siege (attrition applies after you end your turn)
• Your army is besieged by an enemy (attrition applies when your turn starts)
• Your army is currently suffering due to certain events (plague, sickness, earthquake, etc.)
• Your food surplus is negative and your army is currently in a province that imports food.
• Your army is currently in terrain it is not used to (e.g. desert, mountains) and not on a road.
• You cannot pay for your army's or fleet's upkeep (happens when a player has no settlements)
• Your fleet is currently in deep water

The screenshot shows 3 different auras, here is what each are and what they mean:

The character to the right only has one aura, which is red: the red aura only appears on characters that have the red skull above it, which represents attrition damage, see this answer.

The character to the left has three different auras, one of which is barely visible (notice the red aura at the unit's feet). The white aura replacing the red one, which is surrounded by Rays of God only appears on characters which you are currently inspecting by left clicking on it.

The last visible aura in the screenshot is the white circle of light at the unit's feet. It only appears on units that are currently in the "Forced March!" stance, allowing them to move twice the usual distance on the overworld map at the cost of not being able to attack (they can still defend, however), as well as penalties in field of view and morale during combat.
Fleet have the same aura for their own version of "Forced March!", which is called "Double Time!", as well as a similar aura for their "Patrol" stance, which is a white circle of vapor floating above the fleet.