As Marquee executes cues, you have an awful lot of control over how you get from Point A to Point B (whether that be intensity levels or position on stage or from one color to another). Marquee allows you to have independent wait and fade times on intensity levels going up in a cue, levels going down, and on each attribute family (Position, Color, Gobo, Lens and Special attributes). You can also split cues into different Parts giving you more timing options.
Profiles allow you to also affect the method in which a cue goes from Point A to Point B (using the allotted time).
Note:
Apart from simple profiles, Horizon Control's Abstract Control Model gives you greater control on how to execute fades from one Position to another, or from one Color to another using the Pan/Tilt Movement option or the Color Space options.
To assign a profile to a cue (or Cue Part), move the Blue Box to the appropriate timing column (either Cue Time, Down Time, Position, Color, Gobo, Lens or Special) and press [EDIT] or [INSERT]. You can now enter a time (as described in the topic called Blue Box) and/or add a profile to that timing parameter. When the Blue Box is open for editing, the softkeys change to the following:

Pressing any softkey will close the edit box and assign that profile:

If you want to remove a profile, just add a Linear profile to the cue (which is the default).
Note:
[S11] will clear all Independent Timing from the cue. This softkey is only visible when you are editing the Cue Time field.
[S12]will open the Cue Properties dialog box allowing you to add profiles to more than one timing parameter at a time.
Below are images of the available profiles:

This is the default profile. If the profile icon does not appear in the time column, this is the profile used.

Using Snap at Start is equivalent to using a zero time; the attribute goes to the recorded value as soon as the cue is executed.

Using a fade time of 5 seconds with the Snap at End profile is equivalent to assigning a wait time of 5 seconds and a fade time of zero.

The Squares Law profile is a modified power consumption curve that produces a perceived linear output of a dimmed tungsten source. This is generally only used when fading dimmed intensity values up.

The Slow Start curve can be used on dimmers or mechanical attributes of moving lights (such as Position). Apply it when you want to subtly start a change.

Often used when repositioning a moving light, the Slow End very subtly slows and stops the fixture at the end of the fade.

Similar to the EIS curve, the Fast Start is useful when you want to make a real statement at the beginning of a fade.

The Fast End curve can be used as a sort of exclamation mark at the end of a fade; really asserting the final position. Think of a light fading out at the end of a long and dramatic death scene.

Best used for moving light repositioning, the Damped profile gives a more organic movement; more like that of a follow-spot operator. The problem with the Linear fade being used on Pan/Tilt moves is that the fixture is moving at the same rate all the way from point A to point B. The Damped profile slowly starts it up and then slowly parks it at the end. Use it with the Linear Movement function provided by the Abstract Control Model for real human-like moves.

This silly ZigZag Profile provides a sense of humorous indecision when used on an intensity or movement transition.
See Also: