What do the abilities on the hud next to Squadmates represent?

Next to each of my active crew mates is a ability icon:

What do these represent? Is it necessary for me to manage which ability is listed here?

The last ability used. It is generally the ability they will use when you hit the squad member's quick ability key (Q and E on PC by default), unless it is ineffective against the current target, in which case they will use a more appropriate one. If it is greyed out, the ability is cooling down after use. A white flash represents the ability has just cooled down, and a blue icon represents the ability being available.

As near as I can tell, they represent the last active ability that you told your squad member to use.

Besides providing an 'at a glance' way of knowing what your companions cooldown status is, you can quickly activate these abilities by targetting an enemy and hitting Q or E on the PC. On the 360 and PS3, it's the left and right D-pad buttons.