Check, or non-return valves only allow flow in one direction. In the symbol shown on the left, the flow is allowed from the bottom pipe to the top pipe, but not back the other way. There is no spring shown as this valve does not have one; instead, it will rely on gravity to keep it closed.
The valve shown on the right will only allow flow when the pressure at the bottom is higher than the pressure at the top, plus the rating of the spring. Its good practice to write the spring pressure next to the check valve.
Pilot operated check valves have a small pilot line, shown as a dashed line, that is used to lift and open the check valve and allow flow back through the valve. A common format is the double PO check sandwich plate valve, shown below, that is often used with CETOP directional valves. This allows free flow in both directions when pressure is applied to one side but when the directional valve is closed and no pressure is applied then both check valves close, holding the load in place.
Note the small dots placed where the pilot lines and main flow lines meet. These dots are used to highlight the line connections and differentiate from where the two pipes simply cross over each other.
This symbol shows the pilot line entering above the ball. This indicates that pressure will be applied to the back of the ball and therefore close the valve and stop the flow passing through it. The pressure at which the valve will close will depend on the area ratio inside the valve and the relative values of each line pressure.
Shuttle valves are most commonly found on load sensing systems. They are used to always feed the highest pressure to the top connection. The symbol clearly shows that if two different pressures are fed onto the bottom lines then the ball will move either way to allow the maximum pressure onto the top connection.