This may vary slightly from unit to unit but you will have to pass the Pre Joining Fitness Test during your application stage, then work towards the RN Fitness Test levels during New Entry training. When you achieve them you can go to Raleigh/Dartmouth where you have to pass them all again.
Thenafter carry out an annaul 2.4km run or bleep test to stay in date.
Edit: those links don't show it, but there are also some situps (39 for males), pressups (23 for males), a sprint (5x55m-ish in 59secs for males) and a swim test (jump in, tread water for 3 mins, swim 40m, climb out, wearing overalls) to do before and at Raleigh/Dartmouth. No need to pass them again annually though.
Please note: At HMS Raleigh...Swimming Test is jump in deep end of pool, tread water for TWO minutes, swim 40 meters and climb out of the pool unaided.
All with overalls on.......You then have to jump off the high board which is approximately 12 feet high.