In virtually any city there will be a combination of:
- Foreign Bars
- Standing Bars
- Food Bars
- Shot/Cocktail Bars
As explained by uncovery, there are many different chain Irish or British-style pubs. Dubliners, Pig & Whistle, Hub, Hobgoblin, and others. There are also non-chain local versions in less major cities. While these may be designed as 'foreign' pubs, depending on the day of the week and the location, a large portion of the clientele may be Japanese.
Chances are that if you spot a Western resident of Japan between the ages of 20 and 40 in almost any city in Japan and ask, "What bar do foreigners go to?" they will point you to one of these.*1
*1: This may not apply to Kyoto and Tokyo because residents there are typically very different from your average foreign resident of Japan. If in doubt, look up the Hub in these cities, and ask a foreigner or the staff there who can likely direct you to others
These bars tend to have more than one beer on tap, are slightly more pricy than your average Japanese drinkery, and usually have good happy hour specials. They are busiest on weekends, though you may find both Japanese and foreign regulars there on any given weekday.
Standing Bars (立ち飲み)
Standing bars are bars where you stand at the counter and drink. These come in all different varieties. Some will be connected to a liquor store, some will be in a liquor store, some will be in a tiny little hole in the wall, in Fukuoka some will be on the street, and some will just be in a regular bar-type space somewhere. Some even have seats yet still call themselves standing bars.
Virtually all standing bars are on the first floor.
These establishments are usually absurdly cheap, offer happy hour discounts, and are frequented by a marginally older crowd (and usually are a bit more 'downscale'). There are usually regulars, and are typically busier right after work (maybe from 7-9pm) rather than later in the night. In some cities there are standing bars run by women which were all the rage for a while, but are essentially the same thing with a slightly younger crowd, and women tending them instead of men.
Sometimes these will be called 'Food Bars', or 'Cafe/Bar' or something of the sort, but they are places that both offer food and booze, and are essentially multi-use establishments. They offer nice dinners, but after dinner hours have regulars who just drink there. They often have a combination of tables and a counter.
These types of bars are often found slightly off the beaten path, and usually not in the middle of the nightlife. They are often found tucked away on the 3rd floor of some building, and are difficult to identify if you don't speak Japanese. You could walk by them every day and never realize that it was a bar unless someone showed it to you.
This type of bar is the most common type in Japan, especially outside major cities. They are usually not chain establishments, and are run by a single person with a part-timer or two, or a couple. The owner is usually the cook, and these are the true 'locals' in Japan for booze hounds.
The regulars from these bars will often hold barbeques, or go to the beach, or go to an onsen together, etc. They are usually a small community of friends knit around the owner of the bar, and can have very different communities age-wise and interest-wise. I recommend trying to search these ones out. Asking local Japanese people where they drink will likely lead you to one of these places, though if you just ask for a bar they will point you to a foreign bar at first.
The last type of bar is what I'd call a fancy bar. They focus around good liquor and properly mixed cocktails. Usually they are pricier than other types of bars, often have a charge just to sit down, and are a lot mellower than the other types of drinking establishments. Think of them like the bar at a fancy hotel as far as style, only a bit darker and most likely smaller.
These bars usually have a much older clientele (over 40) and strike me as a holdover from the bubble era where people had money to burn, and just never left. They usually have a charge to sit down, but make truly great cocktails. If you are a whiskey-lover, these may be the place for you too, as they often have a wide variety. I usually find these hidden on upper floors of buildings, or tucked away on side streets off the beaten path. If you ask for a shot bar or cocktail bar from someone aged 40+, they should be able to point you in the general direction.
General Tips for Finding Bars/Bar Culture
When you're wandering down a street near the center of a city, and see a sign that lists a price for beer at the top something like this:
You've probably found a bar. Personally at this point I feel no shame in walking in to any establishment, taking a look, and then always asking if there's a charge (「チャージありますか？ Chaaji arimasuka?」). Many bars (depending on region) will charge you just to sit down, or give you a small bit of food in exchange for the charge. If it looks good, and the charge is okay with you, then go ahead and sit down and chat for a bit. Otherwise just give a smile and a nod and walk out. No harm no foul.
When you do sit down with someone who is actually interested in speaking with you (and not just feigning interest to be polite), ask them to show you to another bar. And then you know where to start your adventures the next night. Even if the bar is quiet or nobody is interested in going to another place, the bartenders are usually happy to mark on your map the location of other nearby bars, or recommend general areas to wander.
The one major difference in bar culture is that capacity is limited by the number of seats in most food and cocktail bars. Do not expect to be allowed in if there are no seats open, even if you are willing to stand.
Especially in what I categorized as 'Food Bars', if you go several nights during a couple week span, they will probably remember you even if you show up a couple years later. There is a strong community feel to most local bars that isn't present in a lot of the foreign bars, because the staff doesn't change and the regulars are regular.
Best of luck, and if you tell me what city you're going to I can probably dig you up some places to start your search.