Question: ![The oscilloscope I will be using](https://i.stack.imgur.com/qjAq6.jpg)

At my university, I'll be using the ELENCO S-1325 oscilloscope for a instrumentation lab. Since I'm a mechanical engineering student and not an electrical engineering one, I don't know exactly how to use all the functions though I am familiar with the basic use and operation of it. However, I want to know how to use some of the more complex functions of the oscilloscope.

On an oscilloscope, why would anyone move the variable knobs off their “calibrated” positions? the variable knobs (red VAR knob) for voltage measurements? the small gray “var sweep” knob for the time base?

Isn't it a good thing to keep things calibrated rather than use uncalibrated settings?

Answer: 
If you want to measure the rise or fall time of a digital signal, you use the variable knob for voltage to adjust the amplitude, so that it fits between the horizontal dotted lines (5 divisions). Then you measure the time from 10% to 90% (the middle 4 divisions). Conveniently, there are even percent marks on the screen.

It's less common to use the horizontal variable adjust, maybe it could be used for measuring duty cycle.
Sometimes the relative positions or phases of the signals are more important than the absolute values. E.g. in the photo in your question, you can see that the two traces (i.e. channel 1 and 2) are in phase with each other. If you don't need to know the voltage or exact frequency of the signals, their's no need for exact calibration. 

Also, @starblue's answer is spot-on.

Mmmm - I don't have enough reputation to comment, hence the answer.