I purchased a on demand water heater.my old heater is hooked to a 20 amp breaker(see photo). Do I change out old breaker to a double-pole 60 amp breaker or do I have to move connection to a different breaker?
I don't know what a "double pull" breaker is. There is single and double POLE, and single and double THROW switches (that aren't actually breakers but resemble them - used in transfer switches), but not 'pull'.
The breaker marked HW is the right side of a tandem breaker and the left side is the washer (presumably washing machine not dishwasher). It is, in short, a single POLE double THROW breaker. (Although not really - it's actually two SPST switches on a single breaker I believe)
What this means is that there are two different lines coming off this one breaker, running two different appliances. It is 110V and can never provide more than 110v.
IT IS NOT OK TO REPLACE THIS WITH A 60A.
In order to run a 60A line to an on demand water heater you must put in a 220V, double pole single throw breaker which can feed 220V with 2 hot lines to the new on demand heater.
So yes, you must move to a new spot - and in fact you must move to TWO spots that are adjacent, in order to get the 220v 60A you need.
At a minimum you need a new breaker, which will take up two spaces, and you need to run new wire. The wire that's in place is likely 12 AWG NM, too small for the much higher current the on-demand will draw.
The NEC permits a 60 amp breaker for 6 AWG NM cable; however #6 NM is only rated for 55 amps. If the instructions call for a 60 amp breaker, 6/2 NM is OK as long as the heater only uses 55 amps or less.
Hopefully you've already considered whether your electrical service has adequate capacity for the additional load of the on-demand water heater. Your service was installed to supply a certain maximum capacity, with service conductors and main breaker sized accordingly. If your actual utilization was at or near capacity before, and the increased load of the on demand hot water heater puts you over, it's possible a service upgrade is necessary to supply the increased load.
The cable to your old heater cannot carry the power!
Your existing water heater cable is probably 12/2 (i.e. 12 AWG size wire, black, white, bare-for-ground).
Breakers protect wires. Putting a 60A breaker on that small wire won't do.
The size wire you need for a 60A heater is 6 AWG. 6/2 cable would suffice.
Fitting the breaker
In North America, power is split into two "poles". If you use one pole, you get 120V. If you use both poles, you get 240V.
Generally big loads will switch to 240V before they start bumping the amps, because that's cheaper. So when you see a 30A or 60A load, it's most likely 240V and thus 2-pole.
The existing breaker location can't work. It is a "tandem", "twin" or as we like to call them, "double stuff". You notice it has 2 switches but sits in 1 space in the panel. 1 space only has access to 1 pole. To get both poles, a breaker has to be 2 spaces wide, like the dryer breaker. More on all this in my "what is a double-stuff" answer.
You didn't shoot a pic of your whole panel so I can't see if you have 2 empty spaces adjacent. If you do, you can slap in a common $10 2-pole/60A breaker, e.g. A BR260.
Otherwise you'll need to use a piece of double-stuff exotica called a "Quadplex" breaker. This fits in 2 spaces. It gives a 2-pole breaker in the middle, and either another 2-pole breaker along the edges, or two 1-pole breakers along the edges.