Lipo's are great when you need high capacity, high current and small volume and light weight power. They DO require specialised charging so special charge circuits need to be used. They benefit from charging very quickly - typically 1 hour if you can supply the current required, however there is the danger that they can catch fire during charging, hence the availability of fire resistant charging bags. More specialised Lipo's can be charged in even shorter times, which will be indicated on the battery. Some, more expensive, lipo's have overcharge protection built in.
Most Lipo's are also easily damaged mechanically as they are not in a hard case(some exceptions, but not many) and the device they are used in should provide a safe case.
The safer battery is a lithium ion battery(typically used in mobile phones, laptops etc), but it lacks some of the desirable qualities of the lipo batteries.