The Boss RC-300 supports up to 3 inputs which can be independently controlled. Each channel can be recorded, stopped, started separately, etc., which is then routed to a signal output. The RC-300 is only one example of this type of looper, but in general you will be looking at a higher end product to allow multiple signals/loops to controlled independently.
If you are routing the output signal to a guitar amplifier, you likely won't be happy with the vocal sound. If the output is leading to a mixer for a PA, then a DI box will be necessary between the RC-300 output and the mixer input.
Depending on the level of complexity of what you want to achieve, Ableton Live provides very powerful live performance looping and is widely used by professionals. This solution requires a computer to run the software, and an interface to connect your microphone and instruments. Albeton can then be controlled via a MIDI controller such as a keyboard or sample pad.