The current is the maximum current the charge controller will handle, i.e the maximum the panel can deliver and the maximum you can charge the battery with. With a 50A controller, you can run a 50A panel (at 16V, that's roundabout 800 Watts) and will need roundabout 500 Ah battery capacity.
You could connect the inverter to the DC output of the controller, but then you'd be limited to 360 Watts into the inverter (or roundabout 300 Watts out of it). If you connect the inverter directly to the battery, you'll lose the deep discharge protection from the controller - but that won't matter as any inverter I've seen in the last few years has its own deep discharge cutoff.
And one word of caution: 50 Amps is enough for welding, or burning off your installation. Use sufficiently thick cables (10 mm² would be the absolute minimum - for the metrically challenged: 7 ga or less) and good quality crimp terminals on the ends that you screw into the controller (and appropriate terminals on every other connection).