Mary, firstly to reassure you, not all downloading is illegal
There are lots of legal sites that allow you to download music. Most sell music by signed artists - chart music, albums etc; usually you buy credits and then exchange them for the tracks you want
There are also free downloads; either teasers of upcoming releases by signed artists, or less well known artists, who *give* their music away in the hope that they will be signed by a label.
As for file formats - there are lots, and it depends how you want to use the music; it helps if you have software that plays and coverts files - you often get these free from the same site as the music, or when you buy an MP3 player.
If your niece is downloading chart/popular music for free - then it might be from a file sharing site; these have come under scrutiny from the law, as it is illegal to distribute copywritten material. So if your niece is giving it to you and keeping a copy for herself then she is technically breaking the law, too
Thers lots on the web about downloading music try
about.com look at the *essentials* articles on the lhs as a starting point
Good luck !