The first column is the bare vowels. There are five vowels, and they're pronounced almost the same as the five Spanish vowels. The order of the Japanese vowels are different, however. They are あ、い、う、え、お (a, i, u, e, o), pronounced ah, ee, oo, eh, oh.
Each of the following columns adds consonants in front of these vowels to create a new syllable, each represented by another hiragana character. There's the k-column, か、き、く、け、こ (ka, ki, ku, ke, ko). There are also the s-, t-, n-, h-, m-, and r- columns. There are some exceptions to what you might expect pronunciation-wise in these. Instead of si, there is shi in the s-column. Chi and tsu are where ti and tu would be expected, and ふ in the hu can be pronounced either fu or hu. The consonants are pronounced very much like those in English, but r is the exception.
There's a website called "Engrish Funny" which mocks bad English used in other countries where it's not a primary language. It started mostly mocking the Japanese, and the replacement of the l in English with an r is not just a cute bit of racism. It's actually kind of true. R in Japanese is used for both r and l, and it's really not quite either. The Japanese r is about 75% r, 20% l, and maybe 5% d. This is Koichi's explanation, and it really helps, along with the audio he provides. This set of kana are definitely the hard one to pronounce properly.
In addition to those columns, there are the exceptional y- and w-columns. や、ゆ、よ (ya, yu, yo) are the only three in the y-column, and わ and を (wa and wo) are the extent of the w-column. The y's are pronounced as you'd expect, as is wa, but wo is pronounced the same as o, and that character is only used as a particle to denote the object of a verb.
There's the kind of orphan ん (n) character, which is the only vowelless kana. It's just the n sound.
Beyond these 46 characters, there are 30 dakuten, which are the characters from the k-, s-, t-, and h-columns with a little quote mark next to it, changing the consonant to g, z, d, and b, respectively, or a little circle next to an h-column character to make it a p. Pronunciation exceptions here are ji instead of zi, and dzi and dzu instead of di and du.
Finally there are combinations which include a character with an i (ki, shi, etc.) and one of the y-column characters written smaller next to it. These are pronounced as a single character, like ちゃ is chi+ya=cha, and きょ is ki+yo=kyo.
Overall this gives a huge "alphabet" of 112 possibilities from what amounts to 48 symbols. That's really quite a good range for the effort, I think. Of course, there are 46 other base kana in the katakana "alphabet," but that's another lesson.