How does the Italian "Aperitivo" work?

Note: I am Italian, and travelled my homeland peninsula from north to south, and from west to east.

Sorry to contradict some of the previous answers, but no, aperitivo/apericena are not limited to the Milan area. Maybe it started as a cultural movement in the north of Italy, but I can assure you that nowadays (well, it is since I can remember - that would be at least 25 years) it is widespread across the Italian territory.

How can I find out whether a specific bar offers an aperitivo?

Simple: ask. If you go to aperitivo hours to whatever bar you want, and ask for an aperitivo (including a drink - from beer to cocktail), they will ask you what you want to drink (you can choose from beer to wine, from analcoholic - try Crodino - to a cocktail), and eventually bring you something to eat (chips, peanuts, etc.)
But I guess you want to know where to find good aperitivoes, with better food to eat.

How can I find out which places in the city I am in offer aperitivi?

Places who offer a decent choice of food to eat with your drink do advertise it. Keep an eye for their signs ("Here aperitivo" for example). Ask any Italian if he/she knows a place to get a good aperitivo, they know for sure.

How much should I expect to pay for the drink / aperitivo?

It is common to pay 6-10€ (depending on the place and on the city). In Rome I'd normally pay 7-8€, in Milan I would expect 10€ or more. In the southern part of Italy I would expect 6-7€.

What should I say when ordering to make sure I actually get this aperitivo deal? (In Italian and/or English)

"I would like an aperitivo, please" - "vorrei un aperitivo per cortesia".
You will be asked for the drink that you want. That's it.

What type of drink should I order to blend in with the locals? Can I order any kind of drink to get the aperitivo?

My favourite question! Yes, you can get whatever you want, but make sure you try at least a "Prosecco" (sparkling dry wine) or a "Spritz" (Aperol, Prosecco, sparkling water and a slice of orange).

What type of food can I expect?

This varies a lot from place to place. I've seen bars offering only chips and peanuts to others preparing tiny pizzas ("pizzette"), finger food (olive ascolane, mozzarelline, croquettes, chicken wings), or even complex stuff (pieces of pasta-omelettes, salads, dice of chicken with sauce, cold pasta).

What are the aperitivo hours?

Depending on the part of Italy you are, it can be from 17.00 (south) to 19.00 (north). Apericena is also offered during dinnertimes (a little bit later).

Is it an all-you-can-eat? Will I make a bad impression if I take multiple servings? Will someone stop me and scream at me in Italian?

It is mostly all-you-can-eat. No, you won't make a bad impression, and most likely no one will mind. Especially by apericena, people are expected to eat more, in order not to have a "cena" (dinner).

Is there a difference between "aperitivo" and "apericena"?

"Aperitivo" implies that you are having dinner later (even if after it most people are so full, that they don't), "Apericena" doesn't. My experience is also that apericenas (aperitivo + cena) are offered in a later timespan in comparison to aperitivoes.

How do I avoid tourist traps related to aperitivo?

Ask the locals. I do the same even being an Italian.

What else should I be aware of?

You may be asked to pay in advance. An aperitivo is not shareable (if you are with a friend, and only you order aperitivo, then only you may eat).

The question is rather broad, I'll try to answer each point. Each answer is based on my experience and can be very subjective. Aperitivo experience greatly vary. It's about meeting with friends or colleagues after work, having a couple of drinks which come with something tasty to eat.

How can I find out whether a specific bar offers an aperitivo?

Some places clearly state it on a billboard while others do not. Try to peek customers, whether they receive food with their drinks or not.

How can I find out which places in the city I am in offer aperitivi?

Usual sites may help, better to ask a local, hotel staff or taxi driver.

How much should I expect to pay for the drink / aperitivo?

Prices greatly vary. Under 5 euros you get a drink with some finger food. Between 5-10 euros you can grab whathever you like from the buffet. Take this as a general guideline.

What type of drink should I order to blend in with the locals? Can I order any kind of drink to get the aperitivo?

Aperitivo usually involves alcoholic beverages, as they ease socialization. Typical drinks are spritz Aperol (aka spritz Veneziano), regular wine, beers, negroni. Lately, spritz Hugo (italian only link) is gaining popularity. Ingredients are prosecco wine, sparkling water, Elderflower syrup and decorated with mint leaves.
I wouldn't expect finger food if I order non-alcoholic drinks (this may vary, a buffet aperitivo may have same price for every drink).

What type of food can I expect?

Totally depends on the place. From crappy re-heated leftover pizza to slices of piadina, slices of tramezzino, frittata with truffles, all kinds of pasta (small portions), smoked salmon, shrimps in pink sauce....

What are the aperitivo hours?

I think generally from 6 pm to 9 pm.

Is it an all-you-can-eat? Will I make a bad impression if I take multiple servings? Will someone stop me and scream at me in Italian?

All-you-can-eat aperitivo is more common in big cities like Milan. Again, take a look around and see what the locals are doing. Most probably, no one will tell you anything.
If food is served by a waiter, then it's not a self service buffet. You may ask for more servings, though.

Is there a difference between "aperitivo" and "apericena"?

Aperitivo is the official italian word, go for it. Apericena is a neologism, loaded with poshness. I strongly discourage its use.

How do I avoid tourist traps related to aperitivo?

Central and tourist places are more expensive. If the place is filled with locals, then it's ok. Expect the price to be in the range of 5 - 12 euros.

What else should I be aware of?

Aperitivo is more common in northern Italy and big cities in the center and south (like Rome). Don't expect aperitivi in small villages or hotels' bars.
Drinking age in Italy is 18.

You're asking a lot of questions, and I can answer a few of them.

First, I was under the impression that the aperitivo was something more of the Milan area, but perhaps it has grown into a much wider phenomenon. When I first encountered it, I definitely thought it was a phenomenon.

There does not seem to be an online resource to find out about places offering aperitivo, but Google lists plenty of articles that tell you what they think are the best places for one.

From my experience, the price varies a lot from place to place. 4 euros sounds a bit on the low side to me, and I've seen places charge 8 euros, but the price generally is also indicative of the variety and amount of snacks/buffet on offer.

From my understanding, in principle, when a bar offers aperitivo, which often is limited to certain hours, any drink consumed during that time comes with aperitivo. That said, I've seen some bars offer it explicitly on the menu. Just point, or you can try something like "Questa bevanda viene con aperitivo?"

The drink to get to blend in depends on the bar. Crazy as it may seem, student bars can also offer aperitivo. No surprise, they're immensely popular, with all patrons more typically drinking beer. In other words, drink what you feel like drinking.

The food on offer varies a lot from place to place. I've seen full scale mouthwatering buffets, and 'just' a wide range of salty snacks. As the food is often on display, you can typically check it out before ordering your beverage.

Hours also vary from place to place, but it's fairly typical to run 'before dinner', so starting times could be as early as 4pm, going on to perhaps something like 8pm, though your mileage will vary.

Most likely, no one will scream at you in Italian (or any other language) if you indulge yourself. But, the aperitvo 'works', because virtually no one is a huge glutton while taking part.

I've never seen an apericena. Sounds like it could be more of a meal?

I can imagine that, by now, some aperitivo tourist traps indeed exist. The same applies to all other tourist traps: follow the locals, stay away from the busiest streets.

FWIW My experience in Milan is that it varies from place to place. Sometimes it is complimentary if you are drinking, sometimes you pay extra with your first (or second...) drink to have food as well. The food might become free at the end of the evening if you have been there for a while and there are leftovers.

Sometimes it is a modest mid-evening snack, sometimes there is a big spread from early evening. The food is finger foods and salads, usually there are little plastic party bowls.

Generally the person behind the bar will tell you what the deal is. Drinking Aperol Spritz is probably the way to blend in, but I have had food while drinking beer.

I did notice one woman getting her money's worth by making several trips and totally loading up her bowl each time. Nobody said anything. HTH

The 'buffet' style this article you mention talks about is not common. In general you can expect a plate of small snacks, not a full meal.

The buffets will definitely be more expensive than 4 euro, so I doubt if it'll really be that 'cheap dinner' you talk about. But I guess that depends on how much you eat.