Rendang, Nasi Goreng (Fried Rice), and for those who read SnS, Tempe. But I don't personally recommend the last one unless you fry it in batter real good (Jouichiro's version sounds like something worth trying, though).
Bakso (Meatball soup) is also a common recommendation. It's basically what burgers are for America to us. But if you're asking for authentic Indonesian food, it's actually pretty hard to find one that doesn't scare off tourists imo. Y'see, most of the dishes listed above are actually adaptations of various dishes around the world. Nasi Goreng and Bakso are adaptations from Chinese cuisine, Rendang is an adaptation of Curry, and a lot others I can't remember.
This is due to our archipelago being a massive trading hub in the past, so we've absorbed all of those different kinds of cuisines and adapted it into our own.
The ones we could call authentic are the ones I don't personally recommend, since it's basically a huge hit or miss for some. What you might be looking for would be penyetan or pecel. The former is basically anything edible (meat, fish or whatever) that's pounded and mixed with sambal. The latter is..... I don't even have a clue but basically it's similar to the former except we don't flatten it. Both are typically eaten with rice.
The ones I would really recommend would be our sweets, such as Es Dawet, Es Cendol, Martabak and Surabi. The first two are basically iced desserts, Es Dawet is glutinuous rice flour with a thick brown sugar syrup and rich coconut milk served with ice. Es Cendol is iced grassjelly with sweet syrups and sweetened condensed milk. Martabak and Surabi are our version of pancakes, to put it simply.
Well, that got really wordy. Hope this helps. BTW, did you know Nasi Goreng is a good hangover cure? If you find yourself smashed in Bali or something, your first order of business after waking up would be to order a huge portion of that. Well, actually, I've only heard this from my friend, since I don't drink. So try it out if you like.