Romaji is using Roman (English) letters to represent Japanese sounds. There is a one to one correlation bewteen Hiragana, Katakana & Romaji. Romaji is used for foreigners since a lot of languages use Roman letters. It is also used for commercials and on T-shirts since it is viewed as 'trendy'.

There are three main systems of Romaji. Two (Kunrei-Shiki & Nihon-Shiki) are very similiar in that they tend to classify Romaji according to the patterns as opposed to Hepburn which tends to not follow the pattern in favor of more closely resembling the sounds that each Kana makes. I prefer Hepburn, since it looks like it should sound. Most Japanese people I know use Kunrei-Shiki, since that is what they are taught in schools.

I would recommend to anyone who is serious about learning the Japanese language to only us Romaji for the first little while when you are getting familiar with the sounds and the new words that you are studying. As soon as possible I encourage you to learn Hiragana and when you are ready to buy a dictionary I strongly recommend getting one in Kana (Hiragana & Katakana), relying on Romaji will become a crutch if you use it too long.