There are typically different varieties of timers available in a PLC. There are "On Delay" and "Off Delay" timers. With an "On Delay" timer, the accumulator increments when the timer's input is on and the timer's output (or Done Bit) is set whenever the accumulator equals the preset.
With an "Off Delay" timer, the timer's output (or Done Bit) is set when the input line becomes true, and the accumulator increments when the timer's input returns to an off state. The timer's output (or Done Bit) then returns to an off state when the accumulator equals the preset.
Both of these types of timers reset their accumulator to zero based upon the state of their single control input. With an "On Delay" timer, the accumulator resets when the input is false. With an "Off Delay" timer, the accumulator resets when the input is true.
There are also "Retentive Timers". These timers have two inputs. One input enables incrementation of the accumulator, but does not reset the timer accumulator if the input returns to an off state. The second input is used to reset the timer instead. This allows accumulated timing of intermittent events, while allowing separate reset control based upon a different event or condition.
Hope that helps.