As mentioned, the drive side cup is left-hand threaded, assuming the bottom bracket is English and 68mm. Odds are very good that it is.
The photo of the BB is fuzzy, (close ups require your camera to be set to macro mode) but it looks like you'll need a spline tool similar to Shimano's freewheel tools in order to remove the cups. If the cup(s) have flats, then a conventional BB spanner that fits your cup will work.

If it's an English 68mm BB, then buy your replacement to this spec. How long the square taper spindle should be is determined by the model of crank you select. Pick the crank you want and it should specify the required spindle length.

Good luck!