Footy is a shortened form of the word 'football'.
In Australia, 'footy' and 'football' can mean one of many things. The meaning very much depends on the context in which it is spoken, as well as the origin of the speaker.
In Victoria, where AFL (Australian Football League) is the more popular sport, 'footy' usually means AFL. When my Victorian friend talks about 'footy', I know she means AFL.
In New South Wales, NRL (National Rugby League) is more widely played than AFL, which means that 'footy' often means rugby league, which can also be shortened to 'league'.
A New Zealander might use the nickname 'footy' to mean rugby union, because that form of rugby is more well known in NZ. Rugby union is usually shortened to 'rugby' but can also be called 'union' if one is talking about rugby league ('league') and rugby union ('union') at the same time.
Australians rarely refer to soccer as 'football' unless they have lived in the UK or have European friends and family.
Image credit: http://www.rrepp.com.au/62/blog?category_id=28
