in terms of power supply, there are broadly 2 kinds of opamps -
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single supply (lm358 and similar) - these are generally low power also, they take a +Vcc and GND as power. If what you are amplifying is a symmetrical signal (eg. audio signal), then you will need to capacitively couple and bias it through a resitor voltage divider to somewhere midway in the opamps swing range or the negative portion of your signal will clip... google "biasing"..... and do the same for output. For other uses, this is not necessary.
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dual supply - these take symmetrical +/- Vcc power inputs and your audio signal can be wrt GND directly can be plugged in through a capacitive coupling and the opamp output will swing both positige and negative. however it is a pain to create symmetrical supplies and these opamps often operate at higher voltages
net-net, which opamp u pick depends on use... but for most simple uses, its much easier to work with single supply low power opamps that you can feed power to from arduino's voltage regulator.
On your +/-2.5v question, biasing an input signal will put it at conceptually at gnd... however, i don't really know if there are symmetrical opamps which operate at such low voltages as +/- 2.5 - check the datasheet. But there definitely are single supply low power opamps which work GND to 5V... you simply bias your input signal