The important thing is that you understand that $(form) is the same than jQuery(form).
Constructing a variable in Javascript like that - jQuery('#myThing') or $('#myThing'), then, is the same.
Once you understand that, a step ahead:
Programmers have something that they like to call 'Standards', just to make the code more easy to read. And for the sake of this, the CONVENTIONAL way to SHOW the reader that a variable contains a jQuery variable, and not only a normal Javascript variable, you put a $ in front.
So you have $form = jQuery(form); or the same thing, $form = $(form).
Conventions could have been different, let's say, IAMJQUERYTHING!_form instead $form, but this is how its accepted :)
Another step ahead, by the way:
Writing $(form) makes a jQuery var, same as jQuery(form), but in some cases using some libraries make conflicts appears when using $ instead jQuery. That's why I started using jQuery instead $ in a regular basis.
Hope this helps!