Looking at road bikes and noticed that sometimes there are 2 options, one being compact.
What is the difference? Compact geometry or gearing?!

Compact normally refers to the front gears. which will be 34/50 for compact as opposed to standard 39/53

but some frames get referred to as having compact frame geometry which have sloping toptubes ie Giants

Are there any advantages of having compact gearing?

Climbing a 20% gradient is much easier on a 34×26 gear than a 39×26 gear.

compact gearing is basically lower than normal doubles. normal doubles have 53/39 toothed front chainrings, compacts are 34/49 (i think?) personally i found my compact was too 'spinny' I changed it for a double after about 3 months.