As regards autonomy, the calculation is quite a simple one, even though the thing gets complicated in the cases of “multi voltage”; in general it is enough to operate a division between the global capacity of the battery (expressed in mAh) and the circuit power consumption (expressed in mA) so to obtain the autonomy time (expressed in hours).
For example, an hermetic 12V 2400mAh Lead-acid battery, powering a circuit requiring 12V 300 mA overall, guarantees a maximum autonomy of 2400/300 = 8 hours.
LiPo/Li-Ion batteries have the peculiarity to be able to deliver, even if for a very short time, current for values that are definitely greater than their nominal value, therefore they are very used in fields that require high inrush currents. On the other hand, they do not lend themselves for the creation of the so called “buffer” applications, that is to say, when a circuit is normally powered by the electric network and the batteries, constantly kept under charge, are used only when the power goes out (for example, in domotic installations), since they would get damaged in a very short time.
In these cases, the most suitable batteries are Lead-Acid ones, while NiMh or, worse, the older NiCd, suffer from the so called memory effect, thus it is better to use them until they are fully discharged to recharge them again.
For the usage as buffer, the Lead-acid batteries are definitely advisable, since they can be left constantly under charge, thus lending themselves to the task.
There are however recharge control circuits, very sophisticated ones, that in some cases allow some exceptions to what has been stated above.