"Conbini" (コンビニ) comes from the english word "convenience", of course abbreviated, as the Japanese like so much to do!
A conbini is a kind of store/supermarket that sells a little of everything. It has everything you might need for your daily life. You can buy food, drinks, cleaning and hygienic products, magazines, school material, and a ton of more stuff. Every one of them has an ATM machine, a copy machine, and some even have special machines that allow you to make online payments, buy tickets, charge your mobile, and so on. You can use a toilet for free in every one of them, and not to mention every one of them is open 24 hours a day!
They are really a life saver. But of course the prices might get a little expensive sometimes. After all, it's convenient. So I recommend you to use a normal supermarket where you can get cheaper stuff, and use the conbini only at critic times or if you find some cool discount or point campaign!
There are 6 main conbini stores in Japan (please correct me if there are more), here are their pictures!
I personally think the first three have the best services!