- Romaji (but they don't generally use this, it's just for us gaijin)
When I started out learning Japanese, I had a hard time grappling with this issue. I felt like it was done to spite people learning their language, and wished everything was written in hiragana (I know, wishing everything was written in romaji would have been a better dream, but then I wouldn't have to learn any new alphabet and that would feel too much like cheating). Having been exposed to Japanese for a couple of years now, I feel that I can give an explanation to those who are currently in the position I was in.
Let me tell you first off, having "foreign" words written in katakana is a huge boon to foreigners in Japan. If I see a word in katakana, the first thing I do is try to sound it out and see if it sounds like an English word. It seems that most katakana words are outright English words written using japanese phonetics, however some of them are a kind of pidgin English. Words like so-fa (sofa), gasu (gas), and garasu (glass) are all direct English translations. They basically took the english word, and made a Japanese equivalent that sounds as close to the original English as is possible with their alphabet. A word like pasokon (personal computer) is a hybrid of two english words, but still easier to remember than a completely foreign word. Sure it sucks that you have to learn 48 new katakana symbols that stand for the same thing as 48 new hiragana symbols that you already learned, but it's a small price to pay.
Just have to add this in here to scare people even more. Sometimes there even weirder ways that their alphabets are combined. Take "keshigomu" for instance. The word means "eraser" and is a combination of "keshi" which is basically the root of "to erase" and "gomu" which is katakana for gum. The word actually has kanji, hiragana, and katakana all together... how wonderful right! :)
Now that I've had a chance to read some children's books written soley in hiragana, I can really appreciate why kanji exists. Remember that there are generally no spaces between characters in Japanese. Basically, kanji makes it really easy to figure out where words start and end, similar to spaces in an English sentence. When I only understand maybe 30 percent of a book's vocabulary, I frequently don't know where one word starts and one word ends when they are all written in hiragana. I also tend to find words in between words (where I recognize a word that doesn't exist by taking a combination of the end of one word, and the beginning of another), and my Japanese tutor laughs at me a lot :) With kanji (and furigana, which are little hiragana written above the kanji so I know how it's pronounced), I know for certain where words begin and end.
Another reason why kanji is so important, is because there are several cases of words having the same hiragana as each other, but completely different meanings (but different kanji). Take "hashi" for example. Hashi can mean bridge, it can mean edge or it can mean chopsticks (and a few more things, but you get the point). When written in hiragana, there is no way to tell what the meaning is without context, and even then it can be confusing. When hashi is written in kanji however, the meaning becomes clear.