Finding Bicycle Parts
Generally, parts are somewhat standardized. For example, bottom brackets only come in a few types and different size. 

Given its a recent department store bike, you likely have a standard square taper bottom bracket with ISO threading. You can take the relevant measurements (spindle length, shell width, etc.), but the easiest way for bottom brackets is either read them off the existing bottom bracket (usually labeled) or take it to a bike shop so they can measure it quickly and give you a new one. 

As for other parts, you can usually identify them by what family of parts they belong to if they're in the drivetrain (# of speeds and manufacturer), or measure them for things like seatposts or handlebars (which are also standardized, e.g. road bikes use some sizes, mountain others).