I've done a good bit of reading at places like Wikipedia, but I don't have any background in Jewish history. In layman's terms, could someone tell me what the main differences between Rabbinate and Karaite Judaism? I gather it is somewhat similar to differences between Orthodox/Catholic/Protestant Christianity, but I've really failed to discern the difference in what I have read. Thanks much!
The difference is not quite equivalent to any particular Christian divisions, but is closest to the "sola scriptura" thing some Protestant Groups are into.
The division happened at some point in the Early Middle Ages in the Middle East. Rabbinic Judaism believed (and still believes) that along with the Torah, Moses was given a set of unwritten teachings as well, called the "Oral Law". Over time, this collection of laws was supplemented with Rabbinic ordinances and traditions. At the time, this set of laws was written mostly in the Talmud, along with other texts.
Karaites rejected the Oral Law, and by extension rejected Rabbinic ordinances. They saw them as violating Deuteronomy 13:1 אֵ֣ת כָּל־הַדָּבָ֗ר אֲשֶׁ֤ר אָנֹכִי֙ מְצַוֶּ֣ה אֶתְכֶ֔ם אֹת֥וֹ תִשְׁמְר֖וּ לַעֲשׂ֑וֹת לֹא־תֹסֵ֣ף עָלָ֔יו וְלֹ֥א תִגְרַ֖ע מִמֶּֽנּוּ׃ "Be sure to keep everything I [God] am commanding you today; do not add to it, and do not take away from it" (the "do not add to it" is known in Rabbinic jargon as "bal tosif"). They follow their understanding of the bible, which is not always necessarily literal--there are several commandments Rabbinic Judaism sees as instructing actual physical things, but Karaites see as metaphorical statements.
As a result, they practice Shabbat differently. Rabbinic Jews think the prohibition of fire on the Sabbath is really on kindling a fire, based on both the formulation of a verse in the bible. This allows you to light a fire before and allow it to continue to burn (hence Shabbat candles), whereas Karaites don't think you can have any fire burning on the Sabbath. They also calculate the holiday of Shavuot differently, based on a more straightforward reading of a verse about its date, absent the Rabbinic tradition of how to understand that particular verse.
Nowadays there are very few Karaites. There are some scattered in the US, and a decent size community in Israel. But by the late Middle Ages, the Rabbinic Jews had effectively "won" and the clear majority of Jewish communities were Rabbinic, despite communities with a substantial Karaite minority (Egyptian Karaites in particular, but there were other communities until recently too).