Dual supply opamps will operate on + and - for their supply voltages (e.g. +15V and -15V). Single supply opamps will operate on single supply rail and ground (e.g.+5V and GND).
A dual supply opamp will be able to work with signals that swing above and below ground (e.g. 5V pk-pk sine wave with no DC offset) and will be able to put out positive and negative voltages. A single supply rail will only work with signals above DC ground (e.g. 0-5V square wave) and put out only positive voltage. If the signal goes below ground the lowest a single rail opamp can output is 0V (or however close it can get to ground).
What is "ground" except some random net that you decide to label as such? Single supply and dual supply op-amps are basically the same thing, some are just more "directed" at one situation or another based on their ability to drop down to the negative rail (or ground rail, whatever you want to call it...the lower one!). In fact most rail-to-rail op-amps are labeled as both single and dual-supply, and the spec sheet lists both options.
If you draw up the schematic for an op-amp buffering a 1v signal with 0 and 5v supplies, versus an op-amp buffering a -1.5v signal with -2.5v and +2.5v supplies, you'll see that they're identical, apart from the names you stick on the rails. Since the dual-rail op-amp has no ground reference, as far as it's concerned it IS operating on a single rail, how does it know that you're arbitrarily calling its lower rail "-2.5v"?
As with choosing any part, make sure your operational conditions fall within the part's accepted range. Whether the manufacturer chose to call that particular op-amp a "single" or "dual" supply part makes no difference.