Using romaji is when you write out the Japanese 
language in Roman letters (meaning, the type of 
letters that English is written in - the letters 
you're looking at now). Using kanji (and hiragana 
and katakana) is when you write the Japanese 
language in Japanese characters (basically, the 
things that don't show up unless you have 
Japanese text support on your computer).

Kanji is the way Japanese is *meant* to be 
written. Romaji is really something that was fit 
onto the Japanese language for two main reasons: 
A) To teach the language to foreigners.
B) For computers with 26 character keyboards.

But serious students of Japanese should really 
avoid romaji at all costs since most linguists 
agree that it hinders learning the language.

Many Sharp Smiles,
--Drac