Hi D3V4, and welcome to the SitePoint forums
Both #disclaimer and :visible are selectors
disclaimer basically says "That element in the DOM (e.g. a div) that has id "disclaimer"), and ":visible" indicates if the element is visible.
You could also do $("#disclaimer:visible"), that will return the element with id disclaimer that is visisble. $("disclaimer").is(":visible") returns if the element is visible, and does not return a new jQuery object.
No. The second parameter of the slideUp() function is, like you say, a callback function.
In your first example, $("#hideButton").fadeOut() is only then executed when $("#disclaimer").slideUp("slow") is done (i.e. the animation has completed).
In your second example the two things are executed simultaniously (try both your versions with 4000 instead of "slow" to see this).
No problem, if you have any more questions just ask