Manga Cafes are just what they sound like. They are cafes where you can go and read manga. You pay for the time that you stay there (usually about ¥400 per hour), but you are free to read all the manga you want. Most manga cafes also offer internet access, like the popular internet cafes across Japan. And many internet cafes in Japan also offer manga for reading, making the two mostly interchangeable. Like most of the other types of cafes in Japan, smoking is usually permitted.
Some cafes also offer overnight accommodations at a low price (between ¥2,000 and ¥3,000). Manga (and internet) cafes will give you a cubicle complete with seating, an internet connection, and all the manga you would ever want to read throughout the night. It is kind of neat sleeping in a manga library, but be aware the place will also be full of people who are probably hung over (cafes are a good option for those who have missed their last train, and people who are drunk tend to miss trains). And food is limited to what is available at the vending machines. If you want to check one out, there are many cafes around the country, and your best bet for finding one would be near a train station.