Motion-detection alarm - how to make alarm persist?
You need a latch circuit.



Figure 1. A two-transistor latch circuit. Source: Learning About Electronics.

How it works:


On power-up both transistors are off.
When a positive pulse arrives on the input current flows into the base of the lower transistor - let's call that Q1.
Q1 turns on shorting C to E.
Current is now drawn from the base of Q2 (which is PNP type) so it turns on too.
Current now flows through Q2 lighting the LED.
Current also flows back to Q1's base through the 1k resistor. Now if the original input signal is removed Q2 keeps Q1 on which keeps Q2 on which keeps Q1 on ...


To reset the circuit you just need a normally closed push-button on the positive supply or in the 1k branch.