What kind of horizontal stabilizer does a Boeing 737 have?
A Boeing 737 has a movable horizontal stabilizer for pitch trim with elevators for pitch control (also known as a THS – Trimmable Horizontal Stabilizer). This is true for all 737 variants, including the MAX. Note: The term stabilator is typically used when the entire tail is rotated for pitch control, like e.g. on a Piper Cherokee.

The following image (taken from this question) shows the horizontal stabilizer (the black arc indicates the range of movement of the whole stabilizer):


From the 737 NG FCOMv2 9.20.7 (Flight Controls - System Description):


  Pitch Control
  
  The pitch control surfaces consist of hydraulically powered elevators and an
  electrically powered stabilizer. The elevators are controlled by forward or aft
  movement of the control column. The stabilizer is controlled by autopilot trim or
  manual trim.
  
  Elevators
  
  The elevators provide pitch control around the airplane’s lateral axis. The
  elevators are positioned by the pilots’ control columns. The A and B FLT
  CONTROL switches control hydraulic shutoff valves for the elevators.
  Cables connect the pilots’ control columns to elevator power control units (PCUs)
  which are powered by hydraulic system A and B. The elevators are interconnected
  by a torque tube. With loss of hydraulic system A and B the elevators can be
  mechanically positioned by forward or aft movement of the pilots’ control
  columns. Control forces are higher due to friction and aerodynamic loads.
  
  [...]
  
  Stabilizer
  
  The horizontal stabilizer is positioned by a single electric trim motor controlled
  through either the stab trim switches on the control wheel or autopilot trim. The
  stabilizer may also be positioned by manually rotating the stabilizer trim wheel.
  
  Stabilizer Trim
  
  Stabilizer trim switches on each control wheel actuate the electric trim motor
  through the main electric stabilizer trim circuit when the airplane is flown
  manually. With the autopilot engaged, stabilizer trim is accomplished through the
  autopilot stabilizer trim circuit. The main electric and autopilot stabilizer trim
  have two speed modes: high speed with flaps extended and low speed with flaps
  retracted. If the autopilot is engaged, actuating either pair of stabilizer trim
  switches automatically disengages the autopilot. The stabilizer trim wheels rotate
  whenever electric stabilizer trim is actuated.