DSO Screen Refresh - Start position
The word you are looking for is "trigger".  The trigger system on a scope can seem fairly complex to a beginner.  The trigger system has several controls:

Trigger source: Channel 1, Channel 2, External, Line (AC power line)

Trigger mode: Auto (sweeps runs without a trigger of none available), Normal (sweep only with trigger), Single Sweep (run once on trigger, must be reset for another sweep)

Trigger type: rising edge, falling edge, TV (Analog TV sync)

Trigger Level: a pot to set the voltage at which a trigger occurs.

Usually, on digital scopes, an icon will appear on one side of the screen to indicate the trigger level.

Digital scopes may allow you to set the trigger point to be near the left or right side of the screen, or in the center (pre-trigger or post-trigger).