Japanese version of the english word Convenience Store, usually meaning a small store where food, magazines, and snacks can be bought 24/7. There are various chains active in Japan, amongst the biggest are Lawson, 7-Eleven, Family Mart, SunKus, but with many smaller chains as well.
One of the most convenient things about a konbini (aside from round-the-clock shopping) is that all bills (Gas, Electricity, Telephone, Water) can be paid there, forgoing the need to go to the banks and pay their ridiculous transfer fees.
When in Japan, there is always a konbini around somewhere nearby.