If you are coming to Japan as an ALT then you will need an apartment. Luckily most ALT's will have assistance finding an apartment or won't even have to give it a second thought as either the company you are coming with will organize it for you or the board of education will have an apartment for you.
If you aren't in this group and have to organize an apartment yourself then here are some tips and information to help you know what to expect.
Searching for your first Japanese apartment
Get help from work
If your job has you working in some way for a board of education (kyouikuiinkai/教育委員会), then this would be the first place you want to go (unless you know other people who can help you). If you have any kind of relationship with the people there, they will help you. Find the person who would be most responsible for you and get their advice. They may get someone else to help you, they may help you themselves, they may even have city or teacher housing to offer you.
If you can't do this, then you should head to a local apartment agency. They will have various apartments, just tell them how much your budget is and they will find something to suit.
Find an apartment online
There are many places to look online for an apartment. It's a good way to get an idea of what is available and what you can get for your money but it won't stop you having to eventually get down to a real estate office.
Use the online apartment searching sites to find which real estate offices you want to visit. In my experience, they don't list everything they have so you may find more when you go in.
Try these places:
I'm mostly familiar with the top two as they seemed the easiest to use when I had to use their services.
Generally, make sure you are looking at renting (kariru/借りる) which you might also find shown as rent (chintai/賃貸) then I prefer to search by area. Once you have found the listings by area, then it's usually easy to narrow it down with filters down the side of the page.
Guide to Japanese apartments
Japanese apartment code (to find the right size room)
Apartment codes are a number followed by a letter code. The number indicates how many rooms there are. The lettered code indicates the kitchen size and style. R = Room (as in 1R - there is no kitchen); K = Kitchen (but expect it to be small); DK = Dining Kitchen (kitchen area will be bigger than just K); LDK = Lounge/Large Dining Kitchen (in this case the kitchen and what could have been an adjoining room will be open and counted as one).
eg. A listing for a 3DK means the apartment has 3 rooms as well as a kitchen with room for a table (a small table).
Cost of apartments in Japan
It's really hard to give a Japan wide price here, but for a 1K you can hope for ￥30,000 per month, but may need to pay up to ￥50,000 or more if you want something new and clean. In central Tokyo, ￥100,000 for a 1K isn't ridiculous. And that's just a 1K - kind of like a self-contained dorm room. For a bigger 2K or 2DK outside big cities you can hope for ￥4-50,000 but this will really change depending on location and age of the building.